LOW-DIMENSIONAL AND MULTI-DIMENSIONAL ... - Springer Link

DOI 10.1007/s10958-018-3934-6 Journal of Mathematical Sciences, Vol. 233, No. 3, September, 2018

LOW-DIMENSIONAL AND MULTI-DIMENSIONAL PENDULUMS IN NONCONSERVATIVE FIELDS. PART 2 M. V. Shamolin

UDC 517.9; 531.01

Abstract. In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions. Keywords and phrases: fixed rigid body, pendulum, multi-dimensional body, integrable system, variable dissipation system, transcendental first integral. AMS Subject Classification: 34Cxx, 37E10, 37N05

To the blessed memory and the 100 anniversary of my Grandfather, Guard Captain N. I. Polozov

CONTENTS OF PART 1 (see volume 233, issue 2) Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. Integrability of Equations of Motion of a Pendulum on the Two-Dimensional Plane 2.1. Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Dynamical Equations in the Lie Algebra so(2) . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. First Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Second Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Problem on the Motion of a Free Body under the Action of a Tracing Force . . . . . . . . 2.6. Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity 2.7. Case where the Moment of Nonconservative Forces Depends on the Angular Velocity . . . Chapter 3. Integrability of Equations of Motion of a Pendulum in the Three-Dimensional Space 3.1. Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Set of Dynamical Equations in the Lie Algebra so(3) . . . . . . . . . . . . . . . . . . . . . 3.3. First Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Second Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Problem on the Motion of a Free Body under the Action of a Tracing Force . . . . . . . . 3.6. Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity 3.7. Case where the Moment of Nonconservative Forces Depends on the Angular Velocity . . . Chapter 4. Integrability of Equations of Motion of a Pendulum in the Four-Dimensional Space 4.1. Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Set of Dynamical Equations in the Lie Algebra so(4) . . . . . . . . . . . . . . . . . . . . . 4.3. First Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Second Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174 176 177 178 179 179 180 183 189 194 195 196 197 198 199 204 215 224 224 226 228 229

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 135, Geometry and Mechanics, 2017. c 2018 Springer Science+Business Media, LLC 1072–3374/18/2333–0301 

301

4.5. Problem on the Motion of a Free Body under the Action of a Tracing Force . . . . . . . . 4.6. Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity 4.7. Case where the Moment of Nonconservative Forces Depends on the Angular Velocity . . . Chapter 5. Integrability of Equations of Motion of an an n-Dimensional Rigid Body, n = 5 and n = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. General Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. General Problem on the Motion with a Tracking Force . . . . . . . . . . . . . . . . . . . . 5.3. Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity 5.4. Case where the Moment of Nonconservative Forces Depends on the Angular Velocity . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230 236 250 261 261 264 270 279 288

CONTENTS Chapter 6. Integrability of Equations of Motion of a Pendulum in a Multi-Dimensional Space 6.1. Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Some General Discourses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Set of Dynamical Equations in the Lie Algebra so(n) . . . . . . . . . . . . . . . . . . . . . 6.4. First Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Second Set of Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Problem on the Motion of a Free Body under the Action of a Tracing Force . . . . . . . . 6.7. Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity 6.8. Case where the Moment of Nonconservative Forces Depends on the Angular Velocity . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302 303 304 307 310 311 313 329 368 385

Chapter 6 INTEGRABILITY OF EQUATIONS OF MOTION OF A PENDULUM IN A MULTI-DIMENSIONAL SPACE In this chapter, we systematize some results on the study of the equations of motion of dynamically symmetric multi-dimensional fixed rigid bodies (pendulums) located in nonconservative force fields. The form of these equations is taken from the dynamics of realistic fixed rigid bodies in homogeneous flows of media. We also study the problem on motion of free multi-dimensional rigid bodies in similar force fields, for example, under the action of a nonconservative tracing force. Under the action of such a tracing force, either the magnitude of the velocity of some characteristic point of the body remains constant (this means that the system possesses a nonintegrable servo constraint; see [196, 200, 202, 204]) or the center of mass of the body moves rectilinearly and uniformly (this means that there exists a nonconservative couple of forces in the system; see also [205, 245, 246]). Earlier (see [101]), the author already proved the complete integrability of the equations of a planeparallel motion of a fixed rigid body (pendulum) in a homogeneous flow of a medium under the jet flow conditions when the system of dynamical equations possesses a first integral, which is a transcendental (in the sense of the theory of functions of a complex variable, i.e., it has essential singularities) function of quasi-velocities. It was assumed that the interaction of the medium with the body is concentrated on a part of the surface of the body that has the shape of a (one-dimensional) plate. 302

In [149, 159, 161], the planar problem was generalized to the spatial (three-dimensional) case, where the system of dynamical equations has a complete set of transcendental first integrals. It was assumed that the interaction of a homogeneous medium flow with a fixed body (the spherical pendulum) is concentrated on a part of the body surface that has the shape of a planar (two-dimensional) disk. In the sequel (see [199]), the equations of motion of fixed, dynamically symmetric, four-dimensional rigid bodies, where the force field is concentrated on a part of the body surface that has the shape of a (three-dimensional) disk are studied; in this case the force field is concentrated on a one-dimensional straight line perpendicular to this disk. In this chapter, we present results related to the case where the interaction of the homogeneous flow of a medium with the fixed body is concentrated on a part of the surface of the body that has the shape of an (n − 1)-dimensional disk and the action of the force is concentrated in a direction perpendicular to this disk. These results are systematized and are presented in invariant form. 6.1.

Model Assumptions

Let us consider a homogeneous (n − 1)-dimensional disk D n−1 (with center at the point D) perpendicular to the holder OD in the multi-dimensional Euclidean space En . The disk is rigidly attached perpendicularly to the holder OD located on the (generalized) spherical hinge O in a flow of a homogeneous fluid. In this case, the body is a physical (generalized spherical) pendulum. The medium flow moves from infinity with constant velocity v = v∞ = 0. Assume that the holder does not create a resistance. We assume that the total resistance force S is perpendicular to the disk D n−1 and the application point N of this force is determined by at least the attack angle α between the velocity vector vD of the point D with respect to the flow and the holder OD; the total force is also determined by the direction angles β1 , . . . , βn−2 of the hyperplane of the disk D n−1 (thus, (v, α, β1 , . . . , βn−2 ) are the (generalized) spherical coordinates of the vector vD ), and also the reduced tensor of angular velocity ˜ lΩ , ω ˜∼ = vD

vD = |vD |

˜ is the tensor of angular velocity of the pendulum). Such conditions (l is the length of the holder and Ω generalize the model of streamline flow around spatial bodies (see [59, 86]). The vector OD (6.1.1) e= l determines the orientation of the holder. Then 2 e, S = s(α)vD

(6.1.2)

s(α) = s1 (α) sign cos α,

(6.1.3)

where and the resistance coefficient s1 ≥ 0 depends only on the attack angle α. By the symmetry of the body with respect to the axis Dx1 = OD, the function s(α) is (formally) even. Let Dx1 . . . xn be the coordinate system rigidly attached to the body and, moreover, let the axis Dx1 have a direction vector e, whereas the axes Dx2 , . . . , Dxn−1 , and Dxn lie in the hyperplane of the disk D n−1 . By the angles (ξ, η1 , . . . , ηn−2 ), we define the position of the holder OD in the multi-dimensional space En . In this case, the angle ξ is made by the holder and the direction of the jet flow of the medium. In other words, the angles introduced are the (generalized) spherical coordinates of the point D of the center of a disk D n−1 on the (n − 1)-dimensional sphere of the constant radius OD. 303

The space of positions of this (generalized) spherical (physical) pendulum is the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}, (6.1.4) and its phase space is the tangent bundle of the (n − 1)-dimensional sphere ˙ η˙ 1 , . . . , η˙ n−2 ; ξ, η1 , . . . , ηn−2 ) ∈ R2(n−1) : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. (6.1.5) T∗ Sn−1 {(ξ, ˜ of the angular velocity in the coordinate system Dx1 . . . xn is defined The (second-rank) tensor Ω by a skew-symmetric matrix. For example, for n = 5 this matrix has the form ⎞ ⎛ 0 −ω10 ω9 −ω7 ω4 ⎜ ω10 0 −ω8 ω6 −ω3 ⎟ ⎟ ⎜ ˜ ˜ ⎜ ω8 0 −ω5 ω2 ⎟ (6.1.6) Ω = ⎜ −ω9 ⎟ , Ω ∈ so(5). ⎝ ω7 −ω6 ω5 0 −ω1 ⎠ −ω4 ω3 −ω2 ω1 0 The distance from the center D of the disk D n−1 to the center of pressure (the point N ) is  lΩ |rN | = rN = DN α, β1 , . . . , βn−2 , , vD

(6.1.7)

where rN = {0, x2N , . . . , xnN } in the system Dx1 . . . xn (we omit the tilde over Ω). We note, as in lower-dimensional cases, that the model used to describe the effects of fluid flow on fixed pendulum is similar to the model constructed for a free body and takes into account of the rotational derivative of the moment of the forces of medium influence with respect to the tensor of angular velocity of the pendulum (see also [199]). An analysis of the problem of the (generalized) spherical (physical) pendulum in a flow allows us to find the qualitative analogies in the dynamics of partially fixed bodies and free multi-dimensional bodies. 6.2.

Some General Discourses

6.2.1. Dynamical symmetries of multi-dimensional rigid bodies. Let an n-dimensional rigid body Θ of mass m with smooth (n − 1)-dimensional boundary ∂Θ be under the influence of a nonconservative force field; this can be interpreted as a motion of the body in a resisting medium that fills up the multi-dimensional domain of Euclidean space En . We assume that the body is dynamically symmetric. In this case, for example, for the fourdimensional body, there are two possibilities to represent its inertia tensor in the case of existence of two independent equations on the principal moments of inertia; i.e., in some coordinate system Dx1 x2 x3 x4 attached to the body, the operator of inertia has either the form diag{I1 , I2 , I2 , I2 }

(6.2.1)

diag{I1 , I1 , I3 , I3 }

(6.2.2)

(the so called case (1–3)), or the form (the case (2–2)). In the first case, the body is dynamically symmetric with respect to the hyperplane Dx2 x3 x4 (in other words, Dx1 is the axis of dynamical symmetry) and, in the second case, the two-dimensional planes Dx1 x2 and Dx3 x4 are the planes of dynamical symmetry of the body. For the five-dimensional body, we study the cases of existence of three independent equations on the principal moments of inertia; i.e., in some coordinate system Dx1 x2 x3 x4 x5 attached to the body, the operator of inertia has either the form diag{I1 , I2 , I2 , I2 , I2 } 304

(6.2.3)

(the case (1–4)), or the form diag{I1 , I1 , I3 , I3 , I3 }

(6.2.4)

(the case (2–3)). In the first case, the body is dynamically symmetric with respect to the hyperplane Dx2 x3 x4 x5 (in other words, Dx1 is the axis of dynamical symmetry) and, in the second case, the twodimensional plane Dx1 x2 and three-dimensional plane Dx3 x4 are the planes of dynamical symmetry of the body. Respectively, for the n-dimensional body, we also study the cases of existence of n − 1 independent equations on the principal moments of inertia. In this case, [n/2] variants of the forms (6.2.1) and (6.2.2) (or (6.2.3) and (6.2.4)) are possible (here, [x] is the integer part of x). For example, for the five-dimensional body, three cases (1–5), (2–4), and (3–3) are possible. For the case of an n-dimensional rigid body, primarily, we are interested in the case (1–(n − 1)), i.e., when, in a certain coordinate system Dx1 . . . xn attached to the body, the operator of inertia has the form diag{I1 , I2 , . . . , I2 }; (6.2.5)

 n−1

precisely, in the hyperplane Dx2 . . . xn , a body is dynamically symmetric (in other words, Dx1 is the axis of dynamical symmetry). 6.2.2. Dynamics on so(n) and Rn . The configuration space of a free, n-dimensional rigid body is the direct product (6.2.6) Rn × SO(n) of the space Rn , which defines the coordinates of the center of mass of the body, and the rotation group SO(n), which defines the rotations of the body about its center of mass and has dimension n(n + 1) n(n − 1) = . 2 2 Respectively, the dimension of the phase space is equal to n(n + 1). In particular, if Ω is the tensor of angular velocity of an n-dimensional rigid body (it is a secondrank tensor, see [52]), Ω ∈ so(n), then the part of the dynamical equations of motion corresponding to the Lie algebra so(n) has the following form (see [61, 199]): n+

˙ + ΛΩ˙ + [Ω, ΩΛ + ΛΩ] = M, ΩΛ

(6.2.7)

where (6.2.8) Λ = diag{λ1 , . . . , λn }, −I1 + I2 + . . . + In I1 − I2 + I3 + . . . + In , λ2 = ,..., λ1 = 2 2 I1 + . . . + In−2 − In−1 + In I1 + . . . + In−1 − In , λn = , λn−1 = 2 2 M = MF is the natural projection of the moment of external forces F acting on the body in Rn on the natural coordinates of the Lie algebra so(n), and [·, ·] is the commutator in so(n). The skew-symmetric matrix corresponding to this second-rank tensor Ω ∈ so(5) can be represented in the form ⎞ ⎛ 0 −ω10 ω9 −ω7 ω4 ⎜ ω10 0 −ω8 ω6 −ω3 ⎟ ⎟ ⎜ ⎜ −ω9 (6.2.9) ω8 0 −ω5 ω2 ⎟ ⎟ ⎜ ⎝ ω7 −ω6 ω5 0 −ω1 ⎠ −ω4 ω3 −ω2 ω1 0 (see also [14, 38, 58, 83, 212]), where ω1 , ω2 , . . ., ω10 are the components of the tensor of angular velocity corresponding to the projections on the coordinates of the Lie algebra so(5). 305

In this case, obviously, the following relations hold: λi − λj = Ij − Ii

(6.2.10)

for any i, j = 1, . . . , n. For the calculation of the moment of an external force acting on the body, we need to construct the mapping (6.2.11) Rn × Rn −→ so(n), which maps a pair of vectors (DN, F) ∈ Rn × Rn (6.2.12) n n from R × R to an element of the Lie algebra so(n), where DN = {δ1 , δ2 , . . . , δn },

F = {F1 , F2 , . . . , Fn },

(6.2.13)

and F is an external force acting on the body (here, DN is the vector passing through the origin D of the coordinate system Dx1 . . . xn to the point N of application of the force). To this end, we construct the following auxiliary matrix:  δ1 δ2 . . . δn . (6.2.14) F1 F2 . . . Fn All types of second-order alternating minors of this auxiliary matrix are the coordinates of the moment (DN, F) of the force F, and this moment is currently identified with an element of the Lie algebra so(n). Since the ordering the coordinates ω1 , ω2 , . . ., ωf , f = 1, . . . , n(n − 1)/2, has been introduced on the Lie algebra so(n), we can also introduce the same ordering for the calculating of the moment MF = (DN, F) of the force F. Indeed, the first set G1 of coordinates of the desired moment consists of n − 1 alternating minors          δn−2 δn   δn−3 δn    δn−1 δn  δn  n  δ1       , − , + , . . . , (−1)  . + Fn−1 Fn  Fn−2 Fn  Fn−3 Fn  F1 Fn  The second set G2 of coordinates consists of n − 2 alternating minors        δn−3 δn−1   δn−4 δn−1   δn−2 δn−1      , ...,  , − , + + Fn−2 Fn−1  Fn−3 Fn−1  Fn−4 Fn−1 

  δ1 δn−1  F1 Fn−1

n+1 

(−1)

  . 

Continuing, we obtain the final set Gn−1 of coordinates, which consists of a sole minor    δ1 δ2   . + F1 F2  We see that the first minors in any set begin from the sign “+.” The resulting ordered set from n(n − 1)/2 values is called the coordinates of moment (DN, F) of the force F. Dynamical systems studied in the following sections are, generally speaking, not conservative; they are dynamical systems with variable dissipation with zero mean (see [161, 168]). We examine them by direct methods applicable to systems of dynamical equations, namely, the Newton equation, which plays the role of the equation of motion of the center of mass, i.e., the part of the dynamical equations corresponding to the space Rn : (6.2.15) mwC = F, where wC is the acceleration of the center of mass C of the body and m is its mass. Moreover, due to the higher-dimensional Rivals formula (in this case, it can be obtained by the operator method not difficultly) we have the following relations: ˙ (6.2.16) wC = wD + Ω2 DC + EDC, wD = v˙ D + ΩvD , E = Ω, where wD is the acceleration of the point D, F is the external force acting on the body, and E is the tensor of angular acceleration (a second-rank tensor). 306

If the position of a body Θ in the Euclidean space En is determined by functions that are cyclic (in the following sense: the generalized force F and its moment MF = (DN, F) depend only on the generalized velocities (quasi-velocities) and are independent of the position of a body in the space), then the system of equations (6.2.7) and (6.2.15) on the manifold Rn × so(n) is a closed system of dynamical equations of the motion of a free multi-dimensional rigid body under the action of an external force F. This system has been separated from the kinematic part of the equations of motion on the manifold (6.2.6) and can be examined independently. In particular, the right-hand side of the system (6.2.7) for n = 5 has the form  M = {M1 , M2 , . . . , M10 } = δ4 F5 − δ5 F4 , δ5 F3 − δ3 F5 , δ2 F5 − δ5 F2 , δ5 F1 − δ1 F5 ,  δ3 F4 − δ4 F3 , δ4 F2 − δ2 F4 , δ1 F4 − δ4 F1 , δ2 F3 − δ3 F2 , δ3 F1 − δ1 F3 , δ1 F2 − δ2 F1 , (6.2.17) where M1 , M2 , . . ., M10 are the components of projections on the coordinates in the Lie algebra ⎛ 0 −M10 ⎜ M10 0 ⎜ −M M M =⎜ 9 8 ⎜ ⎝ M7 −M6 −M4 M3 6.3.

the tensor of moment of the external forces in the so(5), ⎞ M9 −M7 M4 −M8 M6 −M3 ⎟ ⎟ (6.2.18) 0 −M5 M2 ⎟ ⎟. M5 0 −M1 ⎠ −M2 M1 0

Set of Dynamical Equations in the Lie Algebra so(n)

In the case of a fixed pendulum, the condition (6.2.5) is realized. Then the dynamical part of the equations of motion corresponding to the Lie algebra so(n), has the following form: (I1 + (n − 3)I2 )ω˙ 1 = 0, ........................... (I1 + (n − 3)I2 )ω˙ r1 −1 = 0, n+1

(n − 2)I2 ω˙ r1 + (−1)

n



(I1 − I2 )Wn−1 (Ω) = (−1) xnN

Ω α, β1 , . . . , βn−2 , v



s(α)v 2 ,

(I1 + (n − 3)I2 )ω˙ r1 +1 = 0, ........................... (I1 + (n − 3)I2 )ω˙ r2 −1 = 0, n

n−1

(n − 2)I2 ω˙ r2 + (−1) (I1 − I2 )Wn−2 (Ω) = (−1)

 xn−1,N

Ω α, β1 , . . . , βn−2 , v



s(α)v 2 ,

(6.3.1)

(I1 + (n − 3)I2 )ω˙ r2 +1 = 0, ........................... (I1 + (n − 3)I2 )ω˙ rn−2 −1 = 0, (n − 2)I2 ω˙ rn−2 (n − 2)I2 ω˙ rn−1

 Ω s(α)v 2 , + (I1 − I2 )W2 (Ω) = −x3N α, β1 , . . . , βn−2 , v  Ω s(α)v 2 , + (I2 − I1 )W1 (Ω) = x2N α, β1 , . . . , βn−2 , v

where rn−2 +1 = rn−1 , and the functions Wt (Ω), t = 1, . . . , n−1, are quadratic forms of the components ω1 , . . . , ωf , f = n(n − 1)/2, of the tensor Ω; moreover, Wt (Ω)|ωk1 =...=ωks =0 = 0,

s = (n − 1)(n − 2)/2,

kj = ri ,

j = 1, . . . , s,

i = 1, . . . , n − 1. (6.3.2) 307

Let us explain the formula (6.3.2). The tensor Ω ∈ so(n) has f = n(n − 1)/2 components. Respectively, the moment of the forces MF = (DN, F) has the same number of components. Since the auxiliary matrix (6.2.14) has the form  0 x2N . . . xnN , (6.3.3) 2 −s(α)vD 0 ... 0 on the right-hand side of the system (6.3.1) at least s = (n − 1)(n − 2)/2 equations contain the identical zero. We denote the numbers of these equations by k1 , . . . , ks . In this case, the corresponding components ωkj , j = 1, . . . , s, of the tensor Ω of the angular velocity are said to be cyclic. The rest of the equations containing the values  Ω s(α)v 2 , l = 2, . . . , n, xlN α, β1 , . . . , βn−2 , v we mark by the indices r1 , . . . , rn−1 since f −s =

n(n − 1) (n − 1)(n − 2) − = n − 1. 2 2

Obviously, Wt (0) ≡ 0 for any t = 1, . . . , n − 1, i.e., the quadratic forms Wt (Ω) are equal to zero identically when all the components of the tensor Ω are equal to zero. In this case, the formula (6.3.2) means that for the vanishing of quadratic forms Wt (Ω), t = 1, . . . , n − 1, it suffices that all the cyclic components of the tensor Ω are zero. In particular, in the case n = 5 this system has the form (I1 + 2I2 )ω˙ 1 = 0, (I1 + 2I2 )ω˙ 2 = 0, (I1 + 2I2 )ω˙ 3 = 0,



3I2 ω˙ 4 + (I1 − I2 )(ω3 ω10 + ω2 ω9 + ω1 ω7 ) = −x5N

Ω α, β1 , β2 , β3 , v



s(α)v 2 ,

(I1 + 2I2 )ω˙ 5 = 0, (I1 + 2I2 )ω˙ 6 = 0, 3I2 ω˙ 7 + (I2 − I1 )(ω1 ω4 − ω6 ω10 − ω5 ω9 ) = x4N (I1 + 2I2 )ω˙ 8 = 0,



Ω α, β1 , β2 , β3 , v



(6.3.4) 2

s(α)v ,

Ω s(α)v 2 , 3I2 ω˙ 9 + (I1 − I2 )(ω8 ω10 − ω5 ω7 − ω2 ω4 ) = −x3N α, β1 , β2 , β3 , v  Ω s(α)v 2 , 3I2 ω˙ 10 + (I2 − I1 )(ω8 ω9 + ω6 ω7 + ω3 ω4 ) = x2N α, β1 , β2 , β3 , v since the moment of the medium’s interaction force for n = 5 is determined by the following auxiliary matrix:  0 x2N x3N x4N x5N , (6.3.5) 2 −s(α)vD 0 0 0 0 where 2 , 0, 0, 0, 0} {−s(α)vD is the decomposition of the force S of the medium’s interaction in the coordinate system Dx1 x2 x3 x4 x5 . In this case, r1 = 4, r2 = 7, r3 = 9, r4 = 10. 308



Since the dimension of the Lie algebra so(n) is equal to f = n(n − 1)/2, the system of equations (6.3.1) represents the set of the dynamical equations on so(n). We see that the right-hand side of Eq. (6.3.1) includes the angles α, β1 , . . . , βn−2 ; therefore, this system of equations is not closed. In order to obtain a complete system of equations of motion of the pendulum, it is necessary to attach several sets of kinematic equations to the dynamic equations on the Lie algebra so(n). 6.3.1. Cyclic first integrals. We note that the system (6.3.1) obtained from (6.2.7), by the existing dynamic symmetry (6.3.6) I2 = . . . = In , possesses s = (n − 1)(n − 2)/2 cyclic first integrals ωk1 ≡ ωk01 = const,

...,

ωks ≡ ωk0s = const,

s=

(n − 1)(n − 2) . 2

(6.3.7)

In this case, we consider the dynamics of our system at zero levels: ωk01 = . . . = ωk0s = 0.

(6.3.8)

In particular, the system (6.3.4) possesses the first integrals ω1 ≡ ω10 ,

ω2 ≡ ω20 ,

ω3 ≡ ω30 ,

ω5 ≡ ω50 ,

ω6 ≡ ω60 ,

ω8 ≡ ω80 ,

(6.3.9)

which are considered at zero levels: ω10 = ω20 = ω30 = ω50 = ω60 = ω80 = 0.

(6.3.10)

The number of nonzero components ωr1 , . . . , ωrp of the tensor Ω is p = f − s = n − 1; here r1 , . . . , rp are the p numbers of the set Q1 = {1, 2, . . . , n(n − 1)/2} that are not equal to k1 , . . . , ks ). Under the conditions (6.3.6)–(6.3.8), the system (6.3.1) has the form of an unclosed system of n − 1 equations:  Ω n s(α)v 2 , (n − 2)I2 ω˙ r1 = (−1) xnN α, β1 , . . . , βn−2 , v ......................................................  Ω s(α)v 2 , (n − 2)I2 ω˙ r2 = (−1)n−1 xn−1,N α, β1 , . . . , βn−2 , v (6.3.11) ......................................................  Ω s(α)v 2 , (n − 2)I2 ω˙ rn−2 = −x3N α, β1 , . . . , βn−2 , v  Ω s(α)v 2 . (n − 2)I2 ω˙ rn−1 = x2N α, β1 , . . . , βn−2 , v In particular, under the conditions (6.3.9)–(6.3.10), the system (6.3.4) has the form of an unclosed system of four equations:  Ω s(α)v 2 , 3I2 ω˙ 4 = −x5N α, β1 , β2 , β3 , v  Ω s(α)v 2 , 3I2 ω˙ 7 = x4N α, β1 , β2 , β3 , v  (6.3.12) Ω s(α)v 2 , 3I2 ω˙ 9 = −x3N α, β1 , β2 , β3 , v  Ω s(α)v 2 . 3I2 ω˙ 10 = x2N α, β1 , β2 , β3 , v 309

6.4.

First Set of Kinematic Equations

In order to obtain a complete system of equations of motion, we need a set of kinematic equations that relate the velocity of the point D (i.e., the center of the disk D n−1 ) and the velocity of the jet flow of medium: ⎛ ⎞ l ⎜ 0 ⎟ ⎟ ˜⎜ (6.4.1) vD = vD · iv (α, β1 , . . . , βn−2 ) = Ω ⎜ .. ⎟ + (−v∞ )iv (−ξ, η1 , . . . , ηn−2 ), ⎝ . ⎠ 0 where

⎛ ⎜ ⎜ ⎜ iv (α, β1 , . . . , βn−2 ) = ⎜ ⎜ ⎜ ⎝

cos α sin α cos β1 sin α sin β1 cos β2 ......... sin α sin β1 . . . sin βn−3 cos βn−2 sin α sin β1 . . . sin βn−2

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(6.4.2)

Equation (6.4.1) expresses the theorem of addition of velocities in projections on the attached coordinate system Dx1 . . . xn . Indeed, the left-hand side of Eq. (6.4.1) is the velocity of the point D of the pendulum with respect to the flow in the projections on the coordinate system Dx1 . . . xn related to the pendulum. Moreover, the vector iv (α, β1 , . . . , βn−2 ) is the unit vector along the axis of the vector vD . The vector iv (α, β1 , . . . , βn−2 ) has the (generalized) spherical coordinates (1, α, β1 , . . . , βn−2 ) which determines the decomposition (6.4.2). The right-hand side of Eq. (6.4.1) is the sum of the velocities of the point D when the pendulum rotates (the first term) and the motion of the flow (the second term). In this case, in the first term, we have the coordinates of the vector OD = {l, 0, . . . , 0} in the coordinate system Dx1 . . . xn . We explain the second term of the right-hand side of Eq. (6.4.1) in more detail. Here (−v∞ ) = {−v∞ , 0, . . . , 0} are the coordinates of the vector in the immovable space. In order to describe it in the projections on the attached coordinate system Dx1 . . . xn , we need to make a (reverse) rotation of the pendulum at the angle (−ξ), which is algebraically equivalent to multiplying the value (−v∞ ) by the vector iv (−ξ, η1 , . . . , ηn−2 ). Thus, the first set of kinematic equations (6.4.1) has the following form: vD cos α = −v∞ cos ξ, vD sin α cos β1 = lωrn−1 + v∞ sin ξ cos η1 , vD sin α sin β1 cos β2 = −lωrn−2 + v∞ sin ξ sin η1 cos η2 , ..........................................

(6.4.3)

vD sin α sin β1 . . . sin βn−3 cos βn−2 = (−1)n+1 lωr2 + v∞ sin ξ sin η1 . . . sin ηn−3 cos ηn−2 , vD sin α sin β1 . . . sin βn−2 = (−1)n lωr1 + v∞ sin ξ sin η1 . . . sin ηn−2 . In particular, in the case n = 5, this set of equations has the following form: vD cos α = −v∞ cos ξ, vD sin α cos β1 = lω10 + v∞ sin ξ cos η1 , vD sin α sin β1 cos β2 = −lω9 + v∞ sin ξ sin η1 cos η2 , vD sin α sin β1 sin β2 cos β3 = lω7 + v∞ sin ξ sin η1 sin η2 cos η3 , vD sin α sin β1 sin β2 sin β3 = −lω4 + v∞ sin ξ sin η1 sin η2 sin η3 . 310

(6.4.4)

6.5.

Second Set of Kinematic Equations

˜ and the We also need a set of kinematic equations that relates the tensor of angular velocity Ω ˙ η˙ 1 , . . ., η˙ n−2 , ξ, η1 , . . ., ηn−2 of the phase space (6.1.5) of pendulum studied, i.e., the coordinates ξ, ˙ η˙ 1 , . . . , η˙ n−2 ; ξ, η1 , . . . , ηn−2 }. tangent bundle T∗ Sn {ξ, Our reasonings are valid for arbitrary dimension. The desired equations are obtained from the following two sets of relations. Since the body moves in the Euclidean space En , we first express the tuple consisting of a phase variables ωr1 , ωr2 , . . . , ωrn−1 , through the new variable z1 , . . . , zn−1 (from the tuple z). For this, we draw the following turns by the angles η1 , . . . , ηn−2 : ⎞ ⎞ ⎛ ⎛ z1 ω r1 ⎟ ⎜ ⎜ ω r2 ⎟ ⎟ = T1,2 (ηn−2 ) ◦ T2,3 (ηn−3 ) ◦ . . . ◦ Tn−2,n−1 (η1 ) ⎜ z2 ⎟ , ⎜ (6.5.1) ⎝ ... ⎠ ⎝ ... ⎠ ωrn−1 zn−1 where the matrix Tk,k+1 (η), k = 1, . . . , n − 2, is obtained from the unit one by the existence of the second-order minor Mk,k+1 : ⎛ ⎞ 1 0 0 0 0 ⎜ ⎟ ⎜ 0 ... ⎟ 0 0 0 ⎜ ⎟ ⎜ (6.5.2) Tk,k+1 = ⎜ 0 0 Mk,k+1 0 0 ⎟ ⎟, ⎜ ⎟ . .. 0 ⎠ ⎝ 0 0 0 0 0 0 0 1  Mk,k+1 =

mk,k mk,k+1 mk+1,k mk+1,k+1

,

mk,k = mk+1,k+1 = cos η,

mk+1,k = −mk,k+1 = sin η.

In other words, the following relations hold: ⎛ ⎞ ⎛ ω r1 z1 ⎜ ω r2 ⎜ z2 ⎟ ⎜ ⎟ ⎜ ⎝ . . . ⎠ = Tn−2,n−1 (−η1 ) ◦ Tn−3,n−2 (−η2 ) ◦ . . . ◦ T1,2 (−ηn−2 ) ⎝ . . . zn−1 ωrn−1

⎞ ⎟ ⎟. ⎠

(6.5.3)

In particular, for n = 5 the values ω4 , ω7 , ω9 , and ω10 are transformed through the composition of the following three turns: ⎛ ⎞ ⎞ ⎛ z1 ω4 ⎜ z2 ⎟ ⎜ ω7 ⎟ ⎜ ⎟ ⎟ ⎜ (6.5.4) ⎝ ω9 ⎠ = T1,2 (η3 ) ◦ T2,3 (η2 ) ◦ T3,4 (η1 ) ⎝ z3 ⎠ , ω10 z4 where

⎛

1 ⎜ 0 T3,4 (η) = ⎜ ⎝ 0 0

⎛ ⎞ 0 0 0 1 0 0 ⎜ 0 cos η − sin η ⎟ 1 0 0 ⎟ , T2,3 (η) = ⎜ ⎝ 0 sin η cos η 0 cos η − sin η ⎠ 0 sin η cos η 0 0 0 ⎛ ⎞ cos η − sin η 0 0 ⎜ sin η cos η 0 0 ⎟ ⎟. T1,2 (η) = ⎜ ⎝ 0 0 1 0 ⎠ 0 0 0 1

⎞ 0 0 ⎟ ⎟, 0 ⎠ 1

311

In other words, the following relations hold: ⎛ ⎛ ⎞ z1 ⎜ ⎜ z2 ⎟ ⎜ ⎟ ⎜ ⎝ z3 ⎠ = T3,4 (−η1 ) ◦ T2,3 (−η2 ) ◦ T1,2 (−η3 ) ⎝ z4

⎞ ω4 ω7 ⎟ ⎟, ω9 ⎠ ω10

(6.5.5)

i.e., z1 = ω4 cos η3 + ω7 sin η3 , z2 = (ω7 cos η3 − ω4 sin η3 ) cos η2 + ω9 sin η2 ,

(6.5.6)

z3 = [(−ω7 cos η3 + ω4 sin η3 ) sin η2 + ω9 cos η2 ] cos η1 + ω10 sin η1 , z4 = [(ω7 cos η3 − ω4 sin η3 ) sin η2 − ω9 cos η2 ] sin η1 + ω10 cos η1 . Then we substitute the following relations instead of the variables z: ˙ zn−1 = ξ, sin ξ , cos ξ sin ξ sin η1 , zn−3 = η˙ 2 cos ξ ........................... zn−2 = −η˙ 1

(6.5.7)

sin ξ sin η1 . . . sin ηn−4 , cos ξ sin ξ sin η1 . . . sin ηn−3 . z1 = (−1)n η˙ n−2 cos ξ z2 = (−1)n+1 η˙ n−3

In particular, for n = 5 we have the following formula: ˙ z4 = ξ, sin ξ , cos ξ sin ξ sin η1 , z2 = η˙2 cos ξ sin ξ sin η1 sin η2 . z1 = −η˙3 cos ξ z3 = −η˙1

(6.5.8)

Thus, two sets of Eqs. (6.5.1) and (6.5.7) give the second set of kinematic equations: ⎞ ω r1 ⎜ ω r2 ⎟ ⎟ ⎜ ⎝ . . . ⎠ = T1,2 (ηn−2 ) ◦ T2,3 (ηn−3 ) ◦ . . . ◦ ωrn−1 ⎛

⎛

⎜ ⎜ ⎜ ⎜ ◦ Tn−3,n−2 (η2 )Tn−2,n−1 (η1 ) ⎜ ⎜ ⎜ ⎜ ⎝

312

sin ξ (−1)n η˙ n−2 cos ξ sin η1 . . . sin ηn−3 sin ξ n+1 (−1) η˙ n−3 cos ξ sin η1 . . . sin ηn−4 ........................... sin ξ η˙ 2 cos ξ sin η1 sin ξ −η˙ 1 cos ξ ξ˙

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ . (6.5.9) ⎟ ⎟ ⎟ ⎠

In particular, for n = 5 we have sin ξ cos η1 sin η2 sin η3 − cos ξ sin ξ sin ξ − η˙2 sin η1 cos η2 sin η3 − η˙3 sin η1 sin η2 cos η3 , cos ξ cos ξ sin ξ ω7 = ξ˙ sin η1 sin η2 cos η3 + η˙1 cos η1 sin η2 cos η3 + cos ξ sin ξ sin ξ sin η1 cos η2 cos η3 − η˙3 sin η1 sin η2 sin η3 , + η˙2 cos ξ cos ξ sin ξ sin ξ cos η1 cos η2 + η˙2 sin η1 sin η2 , ω9 = −ξ˙ sin η1 cos η2 − η˙1 cos ξ cos ξ sin ξ sin η1 . ω10 = ξ˙ cos η1 − η˙1 cos ξ ω4 = −ξ˙ sin η1 sin η2 sin η3 − η˙1

(6.5.10)

We see that three sets of the relations (6.3.11), (6.4.3), and (6.5.9) form the closed system of equations. These three sets of equations include the following functions:   Ω Ω , . . . , xnN α, β1 , . . . , βn−2 , , s(α). (6.5.11) x2N α, β1 , . . . , βn−2 , vD vD In this case, the function s depends only on α, whereas the functions x2N , . . . , xnN depend on the angles α and β1 , . . . , βn−2 and on the reduced tensor of angular velocity lΩ/vD . 6.6.

Problem on the Motion of a Free Body under the Action of a Tracing Force

Together with the problem on the motion of a fixed body, we also examine the motion of a free, dynamically symmetric n-dimensional rigid body (in the case (6.2.1)) with the flat frontal end (the (n − 1)-dimensional disk D n−1 ) in the field of resistance forces under the quasi-stationarity conditions [59, 105] with the same model of the medium’s interaction. If (v, α, . . . , βn−2 ) are the (generalized) spherical coordinates of the velocity vector of the center D of disk D n−1 , lying on the axis of symmetry of a body, Ω is the tensor of angular velocity of the body (for the case n = 5, see (6.1.6)) in the coordinate system Dx1 . . . xn attached to the body (in this case, the axis of symmetry CD coincides with the axis Dx1 , where C is the center of mass), and the axes Dx2 , Dx3 , . . . , Dxn lie in the hyperplane of the disk, I1 , I2 , I3 = I2 , . . ., In = I2 , and m are characteristics of inertia and mass, then the dynamical part of the equations of motion in which the tangent forces of the interaction of the body with the medium are absent, has the form ⎛ ⎞ ⎛ ⎞ v cos α v cos α ⎜ ⎟ ⎜ ⎟ v sin α cos β1 v sin α cos β1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ d ⎜ v sin α sin β1 cos β2 v sin α sin β cos β 1 2 ⎟ + Ω⎜ ⎟+ ⎜ ⎟ ⎜ ⎟ ......... ......... dt ⎜ ⎟ ⎜ ⎟ ⎝ v sin α sin β1 . . . sin βn−3 cos βn−2 ⎠ ⎝ v sin α sin β1 . . . sin βn−3 cos βn−2 ⎠ v sin α sin β1 . . . sin βn−2 v sin α sin β1 . . . sin βn−2 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ −σ −σ F1 /m ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ 2⎜ 0 ⎟ ˙⎜ ⎟ ⎜ ⎟ +Ω ⎜ ⎟ + Ω⎜ ... ⎟ = ⎜ ... ⎟, ⎜ ... ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ 0 0 0 (I1 + (n − 3)I2 )ω˙ 1 = 0, 313

...................................................... (I1 + (n − 3)I2 )ω˙ r1 −1 = 0,  Ω n+1 n s(α)v 2 , (n − 2)I2 ω˙ r1 + (−1) (I1 − I2 )Wn−1 (Ω) = (−1) xnN α, β1 , . . . , βn−2 , (6.6.1) v (I1 + (n − 3)I2 )ω˙ r1 +1 = 0, ...................................................... (I1 + (n − 3)I2 )ω˙ r2 −1 = 0,  Ω n n−1 (n − 2)I2 ω˙ r2 + (−1) (I1 − I2 )Wn−2 (Ω) = (−1) xn−1,N α, β1 , . . . , βn−2 , s(α)v 2 , v (I1 + (n − 3)I2 )ω˙ r2 +1 = 0, ...................................................... (I1 + (n − 3)I2 )ω˙ rn−2 −1 = 0,  Ω s(α)v 2 , (n − 2)I2 ω˙ rn−2 + (I1 − I2 )W2 (Ω) = −x3N α, β1 , . . . , βn−2 , v  Ω s(α)v 2 , (n − 2)I2 ω˙ rn−1 + (I2 − I1 )W1 (Ω) = x2N α, β1 , . . . , βn−2 , v where (6.6.2) F1 = −S, S = s(α)v 2 , σ = CD; in this case    Ω Ω , . . . , xnN α, β1 , . . . , βn−2 , (6.6.3) 0, x2N α, β1 , . . . , βn−2 , v v are the coordinates of the point N of application of the force S in the coordinate system Dx1 x2 . . . xn attached to the body, rn−2 + 1 = rn−1 , and the functions Wt (Ω), t = 1, . . . , n − 1, are the quadratic forms of the components ω1 , . . . , ωf , f = n(n − 1)/2, of the tensor Ω; moreover, the properties (6.3.2) hold. For example, in the case n = 5 our system has the form v˙ cos α − αv ˙ sin α − ω10 v sin α cos β1 + ω9 v sin α sin β1 cos β2 − ω7 v sin α sin β1 sin β2 cos β3 + ω4 v sin α sin β1 sin β2 sin β3 2 + σ(ω10 + ω92 + ω72 + ω42 ) =

F1 , m

˙ cos α cos β1 − β˙1 v sin α sin β1 v˙ sin α cos β1 + αv + ω10 v cos α − ω8 v sin α sin β1 cos β2 + ω6 v sin α sin β1 sin β2 cos β3 − ω3 v sin α sin β1 sin β2 sin β3 − σ(ω9 ω8 + ω6 ω7 + ω3 ω4 ) − σ ω˙10 = 0, ˙ cos α sin β1 cos β2 + β˙1 v sin α cos β1 cos β2 v˙ sin α sin β1 cos β2 + αv − β˙2 v sin α sin β1 sin β2 − ω9 v cos α + ω8 v sin α cos β1 − ω5 v sin α sin β1 sin β2 cos β3 + ω2 v sin α sin β1 sin β2 sin β3 − σ(ω8 ω10 − ω5 ω7 − ω2 ω4 ) + σ ω˙9 = 0, ˙ cos α sin β1 sin β2 cos β3 + β˙1 v sin α cos β1 sin β2 cos β3 v˙ sin α sin β1 sin β2 cos β3 + αv + β˙2 v sin α sin β1 cos β2 cos β3 − β˙3 v sin α sin β1 sin β2 sin β3 + ω7 v cos α − ω6 v sin α cos β1 + ω5 v sin α sin β1 cos β2 − ω1 v sin α sin β1 sin β2 sin β3 + σ(ω6 ω10 + ω5 ω9 − ω1 ω4 ) − σ ω˙7 = 0,

314

v˙ sin α sin β1 sin β2 sin β3 + αv ˙ cos α sin β1 sin β2 sin β3 + β˙1 v sin α cos β1 sin β2 sin β3 + β˙2 v sin α sin β1 cos β2 sin β3 + β˙3 v sin α sin β1 sin β2 cos β3 − ω4 v cos α + ω3 v sin α cos β1 − ω2 v sin α sin β1 cos β2 + ω1 v sin α sin β1 sin β2 cos β3 − σ(ω3 ω10 + ω2 ω9 + ω1 ω7 ) + σ ω˙4 = 0, (6.6.4) (I1 + 2I2 )ω˙ 1 = 0, (I1 + 2I2 )ω˙ 2 = 0, (I1 + 2I2 )ω˙ 3 = 0,



3I2 ω˙ 4 + (I1 − I2 )(ω3 ω10 + ω2 ω9 + ω1 ω7 ) = −x5N

Ω α, β1 , β2 , β3 , v

(I1 + 2I2 )ω˙ 5 = 0,



(I1 + 2I2 )ω˙ 6 = 0, 3I2 ω˙ 7 + (I2 − I1 )(ω1 ω4 − ω6 ω10 − ω5 ω9 ) = x4N (I1 + 2I2 )ω˙ 8 = 0,



s(α)v 2 ,

Ω α, β1 , β2 , β3 , v



s(α)v 2 ,

Ω 3I2 ω˙ 9 + (I1 − I2 )(ω8 ω10 − ω5 ω7 − ω2 ω4 ) = −x3N α, β1 , β2 , β3 , s(α)v 2 , v  Ω s(α)v 2 . 3I2 ω˙ 10 + (I2 − I1 )(ω8 ω9 + ω6 ω7 + ω3 ω4 ) = x2N α, β1 , β2 , β3 , v 

The first part of equations of the system (6.6.1) describes the motion of the center of mass in the n-dimensional Euclidean space En in the projections on the coordinate system Dx1 . . . xn . The second part of equations of the system (6.6.1) is obtained from (6.2.7). In particular, the first five equations of the system (6.6.4) describe the motion of the center of mass in the five-dimensional Euclidean space E5 in the projections on the coordinate system Dx1 x2 x3 x4 x5 , whereas the second part (ten equations) of the system (6.6.4) is also obtained from (6.2.7) for n = 5. Thus, the direct product (6.6.5) R1 × Sn−1 × so(n) of the n-dimensional manifold and the Lie algebra so(n) is the phase space of system (6.6.1) of the dynamical equations of the order n(n + 1)/2. Moreover, since the resistance force is independent of the position of the body in a plane, the system (6.6.1) of dynamical equations is separated from the system of kinematic equations and may be studied independently (see also [199]). In particular, the direct product (6.6.6) R1 × S4 × so(5) of the five-dimensional manifold and the Lie algebra so(5) is the phase space of fifteenth-order system (6.6.4) of the dynamical equations. 6.6.1. Cyclic first integrals. We note that the system (6.6.1) partially obtained from (6.2.7) due to the dynamic symmetry (6.6.7) I2 = . . . = In possesses s = (n − 1)(n − 2)/2 cyclic first integrals ωk1 ≡ ωk01 = const,

...,

ωks ≡ ωk0s = const,

s=

(n − 1)(n − 2) . 2

(6.6.8)

In this case, we consider the dynamics of our system at zero levels: ωk01 = . . . = ωk0s = 0.

(6.6.9)

In particular, the system (6.6.4) possesses the first integrals ω1 ≡ ω10 ,

ω2 ≡ ω20 ,

ω3 ≡ ω30 ,

ω5 ≡ ω50 ,

ω6 ≡ ω60 ,

ω8 ≡ ω80 ,

(6.6.10) 315

which are considered at zero levels: ω10 = ω20 = ω30 = ω50 = ω60 = ω80 = 0.

(6.6.11)

The number of nonzero components ωr1 , . . . , ωrp of the tensor Ω is equal to p = f − s = n − 1; here r1 , . . . , rp are the p numbers from the set Q1 = {1, 2, . . . , n(n − 1)/2} that are distinct from k1 , . . . , ks ). 6.6.2. Nonintegrable constraints. If we consider a more general problem on the motion of a body under the action of a certain tracing force T passing through the center of mass and providing the fulfillment of the equality v ≡ const, (6.6.12) during the motion (see also [199]), then Fx in the system (6.6.1) must be replaced by T − s(α)v 2 .

(6.6.13)

As a result of an appropriate choice of the magnitude T of the tracing force, we can achieve the fulfillment of Eq. (6.6.12) during the motion (see [199]). Indeed, if we formally express the value T by virtue of system (6.6.1), we obtain (for cos α = 0): T = Tv (α, β1 , . . . , βn−2 , Ω) = mσ(ωr21 + . . . + ωr2p )    mσ sin α Ω 2 + s(α)v 1 − Γv α, β1 , . . . , βn−2 , , (6.6.14) (n − 2)I2 cos α v where  Ω = |rN | = (rN , iN (β1 , . . . , βn−2 )) Γv α, β1 , . . . , βn−2 , v  n Ω π  xsN α, β1 , . . . , βn−2 , isN (β1 , . . . , βn−2 ). (6.6.15) = 0 · cos + 2 s=2 v Here isN (β1 , . . . , βn−2 ), s = 1, . . . , n (i1N (β1 , . . . , βn−2 ) ≡ 0) are the components of the unit vector on the axis of the vector rN = {0, x2N , . . . , xnN } on the (n − 2)-dimensional sphere Sn−2 {β1 , . . . , βn−2 }, defined by the equation α = π/2 as the equatorial section of corresponding (n − 1)-dimensional sphere Sn−1 {α, β1 , . . . , βn−2 } (defined by Eq. (6.6.12)), namely, ⎞ ⎛ ⎛ ⎞ 0 0 ⎜ i2N (β1 , . . . , βn−2 ) ⎟ ⎜ ⎟ cos β1 ⎟ ⎜ ⎜ ⎟ ⎜ i3N (β1 , . . . , βn−2 ) ⎟ ⎜ ⎟ sin β1 cos β2 ⎟ ⎜ ⎜ ⎟ = iN (β1 , . . . , βn−2 ) = ⎜ ⎟ ⎜ ⎟ ... ... ⎟ ⎜ ⎜ ⎟ ⎝ in−1N (β1 , . . . , βn−2 ) ⎠ ⎝ sin β1 . . . sin βn−3 cos βn−2 ⎠ inN (β1 , . . . , βn−2 ) sin β1 . . . sin βn−2 π  , β1 , . . . , βn−2 (6.6.16) = iv 2 (see (6.4.2)). 6.6.2.1. Reductions of the system. This procedure can be viewed from two standpoints. First, a transformation of the system has occurred in the presence of the tracing (control) force in the system which provides the corresponding class of motions (6.6.12). Second, we can consider this procedure as a procedure that allows one to reduce the order of the system. Indeed, the system (6.6.1) generates an independent system of the order n(n + 1) (n − 1)(n − 2) − − 1 = 2(n − 1) 2 2 of the following form: 316

⎞ v cos α v sin α cos β1 ⎜ ⎟ v sin α cos β1 ⎜ ⎟ ⎜ ⎟ v sin α sin β1 cos β2 ⎜ ⎟ ⎜ ⎟ d ⎜ v sin α sin β cos β 1 2 (1) ⎟+Ω ⎜ ⎟ ......... ⎜ ⎟ ⎜ ⎟ ......... dt ⎝ ⎜ ⎟ ⎠ v sin α sin β1 . . . sin βn−3 cos βn−2 ⎝ v sin α sin β1 . . . sin βn−3 cos βn−2 ⎠ v sin α sin β1 . . . sin βn−2 v sin α sin β1 . . . sin βn−2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −σ −σ 0 ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ (2) ⎜ 0 ⎟ (1) ˙ ⎜ ⎟ ⎜ ⎟ +Ω ⎜ ⎟ + Ω ⎜ ... ⎟ = ⎜ ... ⎟, ⎜ ... ⎟ ⎜ ⎟ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ 0 0 0  Ω n s(α)v 2 , (6.6.17) (n − 2)I2 ω˙ r1 = (−1) xnN α, β1 , . . . , βn−2 , v  Ω n−1 xn−1,N α, β1 , . . . , βn−2 , s(α)v 2 , (n − 2)I2 ω˙ r2 = (−1) v ......................................................  Ω s(α)v 2 , (n − 2)I2 ω˙ rn−2 = −x3N α, β1 , . . . , βn−2 , v  Ω s(α)v 2 , (n − 2)I2 ω˙ rn−1 = x2N α, β1 , . . . , βn−2 , v where the parameter v is supplemented by the constant parameters specified above; moreover, the matrices Ω(1) , Ω(2) of the size (n − 1) × n can be obtained from the matrices Ω and Ω2 , respectively, by removing the first row. In particular, for n = 5 the system (6.6.4), as a result of actions, forms an independent eighth-order system of the following form: αv ˙ cos α cos β1 − β˙1 v sin α sin β1 + ω10 v cos α − σ ω˙10 = 0, ⎛

⎞

⎛

αv ˙ cos α sin β1 cos β2 + β˙1 v sin α cos β1 cos β2 − β˙2 v sin α sin β1 sin β2 − ω9 v cos α + σ ω˙9 = 0, αv ˙ cos α sin β1 sin β2 cos β3 + β˙1 v sin α cos β1 sin β2 cos β3 + β˙2 v sin α sin β1 cos β2 cos β3 − β˙3 v sin α sin β1 sin β2 sin β3 + ω7 v cos α − σ ω˙7 = 0, αv ˙ cos α sin β1 sin β2 sin β3 + β˙1 v sin α cos β1 sin β2 sin β3 + β˙2 v sin α sin β1 cos β2 sin β3 + β˙3 v sin α sin β1 sin β2 cos β3 − ω4 v cos α + σ ω˙4 = 0, (6.6.18) Ω s(α)v 2 , 3I2 ω˙ 4 = −x5N α, β1 , β2 , β3 , v  Ω s(α)v 2 , 3I2 ω˙ 7 = x4N α, β1 , β2 , β3 , v  Ω s(α)v 2 , 3I2 ω˙ 9 = −x3N α, β1 , β2 , β3 , v  Ω s(α)v 2 , 3I2 ω˙ 10 = x2N α, β1 , β2 , β3 , v 

where the parameter v is supplemented by the constant parameters specified above. 317

The system (6.6.17) is equivalent to the system αv ˙ cos α + . . . = 0, β˙1 v sin α + . . . = 0, β˙2 v sin α sin β1 + . . . = 0, ........................... β˙ n−3 v sin α sin β1 . . . sin βn−4 + . . . = 0, β˙ n−2 v sin α sin β1 . . . sin βn−3 + . . . = 0,  v2 Ω n xnN α, β1 , . . . , βn−2 , s(α), ω˙ r1 = (−1) (n − 2)I2 v  v2 Ω n−1 ω˙ r2 = (−1) xn−1,N α, β1 , . . . , βn−2 , s(α), (n − 2)I2 v

(6.6.19)

...........................  v2 Ω x3N α, β1 , . . . , βn−2 , s(α), ω˙ rn−2 = − (n − 2)I2 v  v2 Ω s(α). x2N α, β1 , . . . , βn−2 , ω˙ rn−1 = (n − 2)I2 v In particular, the system (6.6.18) is equivalent to the system αv ˙ cos α + v cos α {ω10 cos β1 + [(ω7 cos β3 − ω4 sin β3 ) sin β2 − ω9 cos β2 ] sin β1 } + σ {−ω˙10 cos β1 + [ω˙9 cos β2 − (ω˙7 cos β3 − ω˙4 sin β3 ) sin β2 ] sin β1 } = 0, β˙1 v sin α + v cos α {[(ω7 cos β3 − ω4 sin β3 ) sin β2 − ω9 cos β2 ] cos β1 − ω10 sin β1 } + σ {[ω˙9 cos β2 − (ω˙7 cos β3 − ω˙4 sin β3 ) sin β2 ] cos β1 + ω˙10 sin β1 } = 0, β˙2 v sin α sin β1 + v cos α {[ω7 cos β3 − ω4 sin β3 ] cos β2 + ω9 sin β2 } + σ {− [ω˙7 cos β3 − ω˙4 sin β3 ] cos β2 − ω˙9 sin β2 } = 0, (6.6.20) β˙3 v sin α sin β1 sin β2 + v cos α {−ω4 cos β3 − ω7 sin β3 } + σ {ω˙4 cos β3 + ω˙7 sin β3 } = 0,  v2 Ω x5N α, β1 , β2 , β3 , s(α), ω˙ 4 = − 3I2 v  v2 Ω x4N α, β1 , β2 , β3 , s(α), ω˙ 7 = 3I2 v  v2 Ω x3N α, β1 , β2 , β3 , s(α), ω˙ 9 = − 3I2 v  v2 Ω s(α). x2N α, β1 , β2 , β3 , ω˙ 10 = 3I2 v 318

6.6.2.2.

New quasi-velocities in the system. We introduce the new quasi-velocities in our system: ⎛ ⎞ ⎞ ⎛ z1 ω r1 ⎜ z2 ⎟ ⎜ ω r2 ⎟ ⎜ ⎟ ⎟ ⎜ (6.6.21) ⎝ . . . ⎠ = T1,2 (βn−2 ) ◦ T2,3 (βn−3 ) ◦ . . . ◦ Tn−2,n−1 (β1 ) ⎝ . . . ⎠ , ωrn−1 zn−1

where the matrices Tk,k+1 (β), k = 1, . . . , n − 2, are obtained from the identity matrix by the existence of the second-order minor Mk,k+1 : ⎛ ⎞ 1 0 0 0 0 ⎜ ⎟ ⎜ 0 ... 0 0 0 ⎟ ⎜ ⎟ ⎟ (6.6.22) Tk,k+1 = ⎜ ⎜ 0 0 Mk,k+1 0 0 ⎟ , ⎜ ⎟ . .. 0 ⎠ ⎝ 0 0 0 0 0 0 0 1  mk,k+1 mk,k , mk,k = mk+1,k+1 = cos β, mk+1,k = −mk,k+1 = sin β. Mk,k+1 = mk+1,k mk+1,k+1 In other words, the following relations hold: ⎛ ⎞ ⎛ ω r1 z1 ⎜ ω r2 ⎜ z2 ⎟ ⎜ ⎟ ⎜ ⎝ . . . ⎠ = Tn−2,n−1 (−β1 ) ◦ Tn−3,n−2 (−β2 ) ◦ . . . ◦ T1,2 (−βn−2 ) ⎝ . . . zn−1 ωrn−1

⎞ ⎟ ⎟. ⎠

(6.6.23)

In particular, for n = 5, the values ω4 , ω7 , ω9 , and ω10 are transformed through the composition of three turns: ⎛ ⎞ ⎞ ⎛ z1 ω4 ⎜ ⎟ ⎜ ω7 ⎟ ⎟ = T1,2 (β3 ) ◦ T2,3 (β2 ) ◦ T3,4 (β1 ) ⎜ z2 ⎟ , ⎜ (6.6.24) ⎝ z3 ⎠ ⎝ ω9 ⎠ ω10 z4 where

⎛

1 0 0 0 ⎜ 0 1 0 0 T3,4 (β) = ⎜ ⎝ 0 0 cos β − sin β 0 0 sin β cos β ⎛ 1 0 0 0 ⎜ 0 cos β − sin β 0 T2,3 (β) = ⎜ ⎝ 0 sin β cos β 0 0 0 0 1 ⎛ cos β − sin β 0 0 ⎜ sin β cos β 0 0 T1,2 (β) = ⎜ ⎝ 0 0 1 0 0 0 0 1

⎞ ⎟ ⎟, ⎠ ⎞ ⎟ ⎟, ⎠ ⎞ ⎟ ⎟. ⎠

In other words, the following relations hold: ⎛ ⎞ ⎛ z1 ⎜ ⎜ z2 ⎟ ⎟ = T3,4 (−β1 ) ◦ T2,3 (−β2 ) ◦ T1,2 (−β3 ) ⎜ ⎜ ⎝ ⎝ z3 ⎠ z4

⎞ ω4 ω7 ⎟ ⎟, ω9 ⎠ ω10

(6.6.25)

319

i.e., z1 = ω4 cos β3 + ω7 sin β3 , z2 = (ω7 cos β3 − ω4 sin β3 ) cos β2 + ω9 sin β2 , z3 = [(−ω7 cos β3 + ω4 sin β3 ) sin β2 + ω9 cos β2 ] cos β1 + ω10 sin β1 ,

(6.6.26)

z4 = [(ω7 cos β3 − ω4 sin β3 ) sin β2 − ω9 cos β2 ] sin β1 + ω10 cos β1 . 6.6.2.3. Systems of the normal form. We can see from (6.6.20) that the system cannot be solved uniquely with respect to α, ˙ β˙1 , β˙2 , β˙3 on the manifold   π O1 = (α, β1 , β2 , β3 , ω4 , ω7 , ω9 , ω10 ) ∈ R8 : α = k, β1 = πl, β2 = πm, k, l, m ∈ Z . (6.6.27) 2 Thus, formally speaking, the uniqueness theorem is violated on manifold (6.6.27). Moreover, the indefiniteness occurs for even k and any l and m because of the degeneration of the spherical coordinates (v, α, β1 , β2 , β3 ), and an obvious violation of the uniqueness theorem for odd k occurs since the first equation of system (6.6.20) is degenerate for this case. This implies that system (6.6.18) outside of the manifold (6.6.27) (and only outside it) is equivalent to the following system:  σv s(α) Ω Γv α, β1 , β2 , β3 , , α˙ = −z4 + 3I2 cos α v  v2 Ω cos α s(α)Γv α, β1 , β2 , β3 , − (z12 + z22 + z32 ) z˙4 = 3I2 v sin α   Ω σv s(α) − z3 Δv,1 α, β1 , β2 , β3 , + 3I2 sin α v    Ω Ω − z1 Δv,3 α, β1 , β2 , β3 , , + z2 Δv,2 α, β1 , β2 , β3 , v v

z˙3 = z3 z4

z˙2 = z2 z4

320

cos α cos α cos β1 + (z12 + z22 ) sin α sin α sin β1    σv s(α) Ω Ω cos β1 z4 Δv,1 α, β1 , β2 , β3 , − z2 Δv,2 α, β1 , β2 , β3 , + 3I2 sin α v v sin β1    Ω cos β1 Ω v2 + z1 Δv,3 α, β1 , β2 , β3 , s(α)Δv,1 α, β1 , β2 , β3 , , − v sin β1 3I2 v cos α cos α cos β1 cos α 1 cos β2 − z2 z3 − z12 sin α sin α sin β1 sin α sin β1 sin β2    σv s(α) Ω cos β1 Δv,2 α, β1 , β2 , β3 , −z4 + z3 + 3I2 sin α v sin β1    Ω 1 cos β2 σv s(α) Δv,3 α, β1 , β2 , β3 , −z1 + 3I2 sin α v sin β1 sin β2  2 Ω v , (6.6.28) s(α)Δv,2 α, β1 , β2 , β3 , + 3I2 v

z˙1 = z1 z4

cos α cos α cos β1 cos α 1 cos β2 + z1 z2 − z1 z3 sin α sin α sin β1 sin α sin β1 sin β2    σv s(α) Ω cos β1 1 cos β2 + + z2 Δv,3 α, β1 , β2 , β3 , z4 − z3 3I2 sin α v sin β1 sin β1 sin β2  2 v Ω − s(α)Δv,3 α, β1 , β2 , β3 , , 3I2 v  cos α σv s(α) Ω ˙ + Δv,1 α, β1 , β2 , β3 , , β1 = z3 sin α 3I2 sin α v  σv Ω cos α s(α) + Δv,2 α, β1 , β2 , β3 , , β˙2 = −z2 sin α sin β1 3I2 sin α sin β1 v  cos α σv s(α) Ω ˙ β3 = z1 + Δv,3 α, β1 , β2 , β3 , , sin α sin β1 sin β2 3I2 sin α sin β1 sin β2 v

where



  Ω π = (rN , iN β1 + , β2 , β3 ), Δv,1 α, β1 , β2 , β3 , v 2     Ω π π = rN , iN , β2 + , β3 , Δv,2 α, β1 , β2 , β3 , v 2 2   Ω π π π = (rN , iN , , β3 + ), Δv,3 α, β1 , β2 , β3 , v 2 2 2

(6.6.29)

and the function Γv (α, β1 , β2 , β3 , Ω/v) can be represented in the form (6.6.15). In the sequel, the dependence on the variables (α, β1 , β2 , β3 , Ω/v) must be treated as the composite dependence on (α, β1 , β2 , β3 , z1 /v, z2 /v, z3 /v, z4 /v) by virtue of (6.6.26). In the general case, we can see that the system (6.6.19) cannot be solved uniquely with respect to α, ˙ β˙1 , . . ., β˙ n−2 on the manifold   π (6.6.30) α = k, β1 = πl1 , . . . , βn−3 = πln−3 , k, l1 , . . . , ln−3 ∈ Z . 2 Thus, formally speaking, the uniqueness theorem is violated on manifold (6.6.30). Moreover, the indefiniteness occurs for even k and any l1 , . . . , ln−3 because of the degeneration of the spherical coordinates (v, α, β1 , . . . , βn−2 ), and an obvious violation of the uniqueness theorem for odd k occurs since the first equation of system (6.6.19) is degenerate for this case. This implies that system (6.6.17) outside of the manifold (6.6.30) (and only outside it) is equivalent to the following system (n > 2):  σv Ω s(α) Γv α, β1 , . . . , βn−2 , , α˙ = −zn−1 + (n − 2)I2 cos α v

z˙n−1 =

 v2 Ω cos α 2 − (z12 + . . . + zn−2 s(α)Γv α, β1 , . . . , βn−2 , ) (n − 2)I2 v sin α n−2   Ω s(α)  σv , (−1)s zn−1−s Δv,s α, β1 , . . . , βn−2 , + (n − 2)I2 sin α v s=1

321

z˙n−2 = zn−2 zn−1

cos α cos β1 cos α 2 ) + (z12 + . . . + zn−3 sin α sin α sin β1   σv Ω s(α) + zn−1 Δv,1 α, β1 , . . . , βn−2 , (n − 2)I2 sin α v   n−2  Ω cos β1 s+1 (−1) zn−1−s Δv,s α, β1 , . . . , βn−2 , + v sin β1 s=2  Ω v2 , − s(α)Δv,1 α, β1 , . . . , βn−2 , (n − 2)I2 v

cos α cos α cos β1 cos α 1 cos β2 2 − zn−3 zn−2 − (z12 + . . . + zn−4 ) sin α sin α sin β1 sin α sin β1 sin β2     σv s(α) Ω cos β1 + Δv,2 α, β1 , . . . , βn−2 , −zn−1 + zn−2 (n − 2)I2 sin α v sin β1   n−2  Ω 1 cos β 2 (−1)s zn−1−s Δv,s α, β1 , . . . , βn−2 , + v sin β1 sin β2 s=3  Ω v2 s(α)Δv,2 α, β1 , . . . , βn−2 , , (6.6.31) + (n − 2)I2 v

z˙n−3 = zn−3 zn−1

................................................................................. z˙1 = β˙ n−2 (−ωr1 sin βn−2 + ωr2 cos βn−2 )

 v2 Ω s(α)Δv,n−2 α, β1 , . . . , βn−2 , + (−1)n (n − 2)I2 v n−2  cos α  cos βs−1 σv s(α) = z1 (−1)s+1 zn−s + (−1)n+1 × sin α s=1 sin β1 . . . sin βs−1 (n − 2)I2 sin α   n−1  Ω cos βs−1 s (−1) zn+1−s × Δv,n−2 α, β1 , . . . , βn−2 , v sin β1 . . . sin βs−1 s=2  v2 Ω n s(α)Δv,n−2 α, β1 , . . . , βn−2 , , + (−1) (n − 2)I2 v

 cos α σv s(α) Ω ˙ + Δv,1 α, β1 , . . . , βn−2 , , β1 = zn−2 sin α (n − 2)I2 sin α v  σv Ω cos α s(α) ˙ + Δv,2 α, β1 , β2 , β3 , , β2 = −zn−3 sin α sin β1 (n − 2)I2 sin α sin β1 v ................................................................................. cos α β˙ n−2 = (−1)n+1 z1 sin α sin β1 . . . sin βn−3  σv Ω s(α) , + Δv,n−2 α, β1 , . . . , βn−2 , (n − 2)I2 sin α sin β1 . . . sin βn−2 v 322

where

  Ω π = (rN , iN β1 + , β2 , . . . , βn−2 ), Δv,1 α, β1 , . . . , βn−2 , v 2    Ω π π = (rN , iN , β2 + , β3 , . . . , βn−2 ), Δv,2 α, β1 , . . . , βn−2 , v 2 2 ............................................................ (6.6.32)    Ω π π π = (rN , iN , . . . , , βn−3 + , βn−2 ), Δv,n−3 α, β1 , . . . , βn−2 , v 2 2 2   Ω π π π = (rN , iN , . . . , , βn−2 + ), Δv,n−2 α, β1 , . . . , βn−2 , v 2 2 2 and the function Γv (α, β1 , . . . , βn−2 , Ω/v) can be represented in the form (6.6.15). In the sequel, the dependence on the variables (α, β1 , . . . , βn−2 , Ω/v) must be treated as the composite dependence on (α, β1 , . . . , βn−2 , z1 /v, . . . , zn−1 /v) by virtue of (6.6.23); in this case (rN , iN ) is the Euclidean scalar product. 

6.6.2.4. Remarks on the distribution of indices. The values Δv,s (α, β1 , . . . , βn−2 , Ω/v), s = 1, . . ., n − 2, are involved on the right-hand side of the system (6.6.31) linearly with the common multiplier s(α) σv (n − 2)I2 cos α (their number is exactly (n − 2)). For example, the second equation of the system (6.6.31) (i.e., the equation with the left-hand side z˙n−1 ) contains all functions (6.6.32) with indices s from 1 to n − 2 (once for every index), i.e., 1 2 3 4 . . . n − 2. (6.6.33) However, the following equations of the system (6.6.31) contain functions (6.6.32) whose numbers are governed by a more difficult rule. For example, the equation for z˙n−2 contains functions (6.6.32) with the indices (6.6.33), whereas the equations for z˙n−3 contain functions with the indices 2 2 3 4 ... n − 2

(6.6.34)

i.e., the function Δv,2 (α, β1 , . . . , βn−2 , Ω/v) is repeated twice. The common distribution of the indices is given in Table 1. Table 1. Distribution of the indices of set of functions (6.6.32) Left-hand side of system (6.6.31)

Distribution of indices s of set of functions (6.6.32)

z˙n−2

1

2

3

4

...

n−2

z˙n−3

2

2

3

4

...

n−2

z˙n−4

3

3

3

4

...

n−2

z˙n−5

4

4

4

4

...

n−2

...

...

...

...

...

...

...

z˙1

n − 2 n − 2 n − 2 n − 2 ...

n−2

For example, the first-order minor (1) in the upper-left corner of Table 1 corresponds to the case n = 3 and indicates the presence of the function (6.6.32) with s = 1 in the dynamical equations. 323

Similarly, the second-order minor



1 2 2 2 corresponds to the case n = 4 and indicates the presence of the functions (6.6.32) with s = 1, 2 in the dynamical equations, whereas the third-order minor ⎛ ⎞ 1 2 3 ⎝ 2 2 3 ⎠ 3 3 3 corresponding to the case n = 5 and indicates the presence of the functions (6.6.32) with s = 1, 2, 3 in the dynamical equations (6.6.31), etc. 6.6.2.5. Violation of the uniqueness theorem. The uniqueness theorem is violated for the system (6.6.19) on the manifold (6.6.30) for odd k in the following sense: regular phase trajectories of system (6.6.19) pass through almost all points of the manifold (6.6.30) for odd k and intersect the manifold (6.6.30) at a right angle, and also there exists a phase trajectory that completely coincides with the specified point at all time instants. However, these trajectories are different since they correspond to different values of the tracing force. Let us prove this assertion. As was shown above, to fulfill constraint (6.6.12), one must choose the value of T for cos α = 0 in the form (6.6.14). Let   s(α) Ω Ω Γv α, β1 , . . . , βn−2 , = L β1 , . . . , βn−2 , . (6.6.35) lim v v α→π/2 cos α Note that |L| < +∞ if and only if       ∂ Ω Γv α, β1 , . . . , βn−2 , s(α)  < +∞. (6.6.36) lim  v α→π/2 ∂α For α = π/2, the necessary magnitude of the tracing force can be found from the equality  π mσLv 2 2 , β1 , . . . , βn−2 , Ω = mσ(ωr21 + . . . + ωn−1 )− , n > 2, (6.6.37) T = Tv 2 (n − 2)I2 where ωr1 , . . . , ωn−1 are arbitrary. On the other hand, if the rotation around a certain point W of the Euclidean plane En is supported by the action of a tracing force, then this tracing force has the form π  mv 2 , β1 , . . . , βn−2 , Ω = , (6.6.38) T = Tv 2 R0 where R0 is the distance CW . Generally speaking, Eqs. (6.6.37) and (6.6.38) define different values of the tracing force T for almost all points of manifold (6.6.30), and the proof is complete. 6.6.3. Constant velocity of the center of mass. If we consider a more general problem on the motion of a body under the action of a certain tracing force T passing through the center of mass and providing the fulfillment of the equality (6.6.39) VC ≡ const during the motion (see also [199], VC is the velocity of the center of mass), then Fx in system (6.6.1) must be replaced by zero since the nonconservative couple of the forces acts on the body: T − s(α)v 2 ≡ 0.

(6.6.40)

Obviously, we must choose the value of the tracing force T as follows: T = Tv (α, β1 , . . . , βn−2 , Ω) = s(α)v 2 , 324

T ≡ −S.

(6.6.41)

The choice (6.6.41) of the magnitude of the tracing force T is a particular case of the possibility of separation of an independent lower-order subsystem after a certain transformation of the system (6.6.1). Indeed, let the following condition hold for T : T = Tv (α, β1 , . . . , βn−2 , Ω)   n−1  Ω Ω ωri ωrj = T1 α, β1 , . . . , βn−2 , v2 , = τi,j α, β1 , . . . , βn−2 , v v

ω0 = v. (6.6.42)

i,j=0, i≤j

First, we introduce the new quasi-velocities (6.6.21)–(6.6.23). We rewrite the system (6.6.1) for the cases (6.6.7)–(6.6.9) in the form n−1    v2 Ω 2 zs cos α − σ s(α) sin α · Γv α, β1 , . . . , βn−2 , v˙ + σ (n − 2)I v 2 s=1   T1 α, β1 , . . . , βn−2 , Ωv v 2 − s(α)v 2 = cos α, m n−1    v2 Ω 2 zs sin α − σ s(α) cos α · Γv α, β1 , . . . , βn−2 , αv ˙ + zn−1 v − σ (n − 2)I2 v s=1   s(α)v 2 − T1 α, β1 , . . . , βn−2 , Ωv v 2 sin α, = m  σv Ω s(α) · Δv,1 α, β1 , . . . , βn−2 , = 0, β˙1 sin α − zn−2 cos α − (n − 2)I2 v  σv Ω ˙ s(α) · Δv,2 α, β1 , . . . , βn−2 , = 0, (6.6.43) β2 sin α sin β1 + zn−3 cos α − (n − 2)I2 v ...................................................  σv Ω s(α) · Δv,n−2 α, β1 , . . . , βn−2 , = 0, β˙ n−2 sin α sin β1 . . . sin βn−3 + (−1)n z1 cos α − (n − 2)I2 v  v2 Ω n s(α), ω˙ r1 = (−1) xnN α, β1 , . . . , βn−2 , (n − 2)I2 v  v2 Ω n+1 x s(α), ω˙ r2 = (−1) α, β1 , . . . , βn−2 , (n − 2)I2 (n−1)N v ...................................................  v2 Ω s(α). x2N α, β1 , . . . , βn−2 , ω˙ rn−1 = (n − 2)I2 v Here, as above, we introduce the functions (6.6.15) and (6.6.32). In the sequel, the dependence on the variables (α, β1 , . . . , βn−2 , Ω/v) must be treated as the composite dependence on (α, β1 , . . . , βn−2 , z1 /v, . . . , zn−1 /v) by virtue of (6.6.23), as previously. If we introduce the new dimensionless phase variables and the differentiation by the formulas  (6.6.44) zk = n1 vZk , k = 1, . . . , n − 1, · = n1 v  , n1 > 0, n1 = const, we see that the system (6.6.43) has the following form: v  = vΨ(α, β1 , . . . , βn−2 , Z),

(6.6.45)

325

α = −Zn−1 + σn1

n−1  s=1

 Zs2 sin α +

σ s(α) cos α · Γv (α, β1 , . . . , βn−2 , n1 Z) (n − 2)I2 n1 −

 Zn−1

T1 (α, β1 , . . . , βn−2 , n1 Z) − s(α) sin α, (6.6.46) mn1

n−2   cos α s(α) = · Γ (α, β , . . . , β , n Z) − Zs2 v 1 n−2 1 2 sin α (n − 2)I2 n1 s=1  n−2 σ s(α)  (−1)s Zn−1−s Δv,s (α, β1 , . . . , βn−2 , n1 Z) + (n − 2)I2 n1 sin α s=1

− Zn−1 · Ψ (α, β1 , . . . , βn−2 , Z) , (6.6.47)  Zn−2

 ×

cos α = Zn−2 Zn−1 + sin α

n−3  s=1

 Zs2

s(α) cos α cos β1 σ + sin α sin β1 (n − 2)I2 n1 sin α

Zn−1 Δv,1 (α, β1 , . . . , βn−2 , n1 Z) +

n−2 

(−1)

s=2

−

 Zn−3

s+1

cos β1 Zn−1−s Δv,s (α, β1 , . . . , βn−2 , n1 Z) sin β1

s(α) · Δv,1 (α, β1 , . . . , βn−2 , n1 Z) − Zn−2 · Ψ (α, β1 , . . . , βn−2 , Z) , (6.6.48) (n − 2)I2 n21

cos α cos α cos β1 = Zn−3 Zn−1 − Zn−3 Zn−2 − sin α sin α sin β1 +



n−4  s=1

 Zs2

cos α 1 cos β2 sin α sin β1 sin β2

  σ s(α) cos β1 {Δv,2 (α, β1 , . . . , βn−2 , n1 Z) −Zn−1 + Zn−2 (n − 2)I2 n1 sin α sin β1 +

n−2 

(−1)s Zn−1−s Δv,s (α, β1 , . . . , βn−2 , n1 Z)

s=3

1 cos β2 } sin β1 sin β2

s(α) · Δv,2 (α, β1 , . . . , βn−2 , n1 Z) − Zn−3 · Ψ (α, β1 , . . . , βn−2 , Z) , (6.6.49) (n − 2)I2 n21 ............................................................ n−2  cos α  cos βs−1  s+1 (−1) Zn−s Z1 = Z1 sin α s=1 sin β1 . . . sin βs−1 n−1   s(α) cos β σ s−1 (−1)n+1 Δv,n−2 (α, β1 , . . . , βn−2 , n1 Z) (−1)s Zn+1−s + (n − 2)I2 n1 sin α sin β1 . . . sin βs−1 +

s=2

+ (−1)n

s(α) Δv,n−2 (α, β1 , . . . , βn−2 , n1 Z) − Z1 · Ψ (α, β1 , . . . , βn−2 , Z) , (6.6.50) (n − 2)I2 n21

σ cos α s(α) + Δv,1 (α, β1 , . . . , βn−2 , n1 Z) , sin α (n − 2)I2 n1 sin α σ cos α s(α) + Δv,2 (α, β1 , . . . , βn−2 , n1 Z) , β2 = −Zn−3 sin α sin β1 (n − 2)I2 n1 sin α sin β1 .....................................................................

β1 = Zn−2

326

(6.6.51) (6.6.52) (6.6.53)

cos α sin α sin β1 . . . sin βn−3 σ s(α) + Δv,n−2 (α, β1 , . . . , βn−2 , n1 Z) , (n − 2)I2 n1 sin α sin β1 . . . sin βn−3

 βn−2 = (−1)n+1 Z1

where Ψ(α, β1 , . . . , βn−2 , Z) = −σn1

n−1  s=1

(6.6.54)

 Zs2 cos α

σ s(α) sin α · Γv (α, β1 , . . . , βn−2 , n1 Z) + (n − 2)I2 n1 T1 (α, β1 , . . . , βn−2 , n1 Z) − s(α) cos α. (6.6.55) + mn1 We see that the independent subsystem (4.5.33)–(4.5.38) of the order 2(n − 1) can be substituted into the system (6.6.45)–(6.6.54) of the order 2(n − 1) + 1 and can be considered separately on its own 2(n − 1)-dimensional phase space, i.e., the tangent bundle T∗ Sn−1 {Zn−1 , . . . , Z1 ; α, β1 , . . . , βn−2 } of the (n − 1)-dimensional sphere Sn−1 {α, β1 , . . . , βn−2 }. In particular, if the condition (6.6.41) holds, then the method of separation of an independent subsystem of the order 2(n − 1) is also applicable. In the sequel, the dependence on the variables (α, β1 , . . . , βn−2 , Ω/v) must be treated as the composite dependence on (α, β1 , . . . , βn−2 , z1 /v, . . . , zn−1 /v) (and, further, on (α, β1 , . . . , βn−2 , n1 Z1 , . . . , n1 Zn−1 )) by virtue of (6.6.23) and (6.6.44). In particular, for n = 5 the system (6.6.45)–(6.6.54) has the following form: v  = vΨ(α, β1 , β2 , β3 , Z),

(6.6.56)

  α = −Z4 + σn1 Z12 + Z22 + Z32 + Z42 sin α σ s(α) cos α · Γv (α, β1 , β2 , β2 , n1 Z) + 3I2 n1 T1 (α, β1 , . . . , βn−2 , n1 Z) − s(α) sin α, (6.6.57) − mn1 Z4 =

 2  s(α) 2 2 cos α · Γ (α, β , β , β , n Z) − Z + Z + Z v 1 2 3 1 1 2 3 sin α 3I2 n21  σ s(α) − Z3 Δv,1 (α, β1 , β2 , β3 , n1 Z) + Z2 Δv,2 (α, β1 , β2 , β3 , n1 Z) + 3I2 n1 sin α  − Z1 Δv,3 (α, β1 , β2 , β3 , n1 Z)

Z3

− Z4 · Ψ (α, β1 , β2 , β3 , Z) , (6.6.58)

  cos α  2 σ s(α) 2 cos α cos β1 + Z1 + Z2 Z4 Δv,1 (α, β1 , β2 , β3 , n1 Z) = Z3 Z4 + sin α sin α sin β1 3I2 n1 sin α  cos β1 cos β1 + Z1 Δv,3 (α, β1 , β2 , β3 , n1 Z) − Z2 Δv,2 (α, β1 , β2 , β3 , n1 Z) sin β1 sin β1 −

s(α) · Δv,1 (α, β1 , β2 , β3 , n1 Z) − Z3 · Ψ (α, β1 , β2 , β3 , Z) , (6.6.59) 3I2 n21

327

Z2 = Z2 Z4

cos α cos α cos β1 cos α 1 cos β2 − Z12 − Z2 Z3 sin α sin α sin β1 sin α sin β1 sin β2   σ s(α) cos β1 + Δv,2 (α, β1 , β2 , β3 , n1 Z) −Z4 + Z3 3I2 n1 sin α sin β1   1 cos β2 σ s(α) Δv,3 (α, β1 , β2 , β3 , n1 Z) −Z1 + 3I2 n1 sin α sin β1 sin β2 s(α) + · Δv,2 (α, β1 , β2 , β3 , n1 Z) − Z2 · Ψ (α, β1 , β2 , β3 , Z) , (6.6.60) 3I2 n21

cos α cos α cos β1 cos α 1 cos β2 − Z1 Z3 + Z1 Z2 sin α sin α sin β1 sin α sin β1 sin β2   cos β1 1 cos β2 σ s(α) + Δv,3 (α, β1 , β2 , β3 , n1 Z) Z4 − Z3 + Z2 3I2 n1 sin α sin β1 sin β1 sin β2 s(α) Δv,3 (α, β1 , β2 , β3 , n1 Z) − Z1 · Ψ (α, β1 , β2 , β3 , Z) , (6.6.61) − 3I2 n21

Z1 = Z1 Z4

σ s(α) cos α + Δv,1 (α, β1 , β2 , β3 , n1 Z) , sin α 3I2 n1 sin α cos α σ s(α) + Δv,2 (α, β1 , β2 , β3 , n1 Z) , β2 = −Z2 sin α sin β1 3I2 n1 sin α sin β1 cos α β3 = Z1 sin α sin β1 sin β2 σ s(α) + Δv,3 (α, β1 , β2 , β3 , n1 Z) , 3I2 n1 sin α sin β1 sin β2

β1 = Z3

(6.6.62) (6.6.63)

(6.6.64)

where   Ψ(α, β1 , β2 , β3 , Z) = −σn1 Z12 + Z22 + Z32 + Z42 cos α σ s(α) sin α · Γv (α, β1 , β2 , β3 , n1 Z) + 3I2 n1 T1 (α, β1 , β2 , β3 , n1 Z) − s(α) + cos α. (6.6.65) mn1 We see that the independent subsystem of the eighth-order (6.6.57)–(6.6.64) can be substituted into the system of the ninth-order (6.6.56)–(6.6.64) and can be considered separately on its own eight-dimensional phase space, i.e., the tangent bundle T∗ S4 {Z4 , Z3 , Z2 , Z1 ; α, β1 , β2 , β3 } of the fourdimensional sphere S4 {α, β1 , β2 , β3 }. In particular, if condition (6.6.41) holds, then the method of separation of an independent eighthorder subsystem is also applicable. 6.6.3.1. Remarks on the distribution of indices. The values Δv,s (α, β1 , . . . , βn−2 , n1 Z), s = 1, . . . , n− 2, are involved on the right-hand side of the system (6.6.46)–(6.6.54) linearly with the common multiplier s(α) σ ; (n − 2)I2 n1 sin α  on the left-hand the number of these values is exactly (n − 2). For example, Eq. (6.6.47) (with Zn−1 side) contains the functions (6.6.32) with indices s from 1 to n − 2, i.e., 1 2 3 4 . . . n − 2. 328

(6.6.66)

However, Eqs. (6.6.48)–(6.6.50) contain the functions (6.6.32) whose numbers are governed by a more  difficult rule. For example, the equation for Zn−2 contains functions (6.6.32) with the indices (6.6.33),  whereas the equations for Zn−3 contain functions with the indices 2 2 3 4 ... n − 2 i.e., the function Δv,2 (α, β1 , . . . , βn−2 , n1 Z) is repeated twice. The common distribution of the indices is given in Table 2. Table 2. Distribution of the indices of set of functions (6.6.32) Left-hand side of system (6.6.46)–(6.6.54)

Distribution of indices s of set of functions (6.6.32)

 Zn−2

1

2

3

4

...

n−2

 Zn−3

2

2

3

4

...

n−2

 Zn−4

3

3

3

4

...

n−2

 Zn−5

4

4

4

4

...

n−2

...

...

...

...

...

...

...

Z1

n − 2 n − 2 n − 2 n − 2 ...

n−2

For instance, the first-order minor (1) in the upper-left corner of Table 2 corresponds to the case n = 3 and indicates the presence of the function (6.6.32) with s = 1 in the dynamical equations. Similarly, the second-order minor  1 2 2 2 corresponds to the case n = 4 and indicates to the presence of the functions (6.6.32) with s = 1, 2 in the dynamical equations, whereas the third-order minor ⎛ ⎞ 1 2 3 ⎝ 2 2 3 ⎠ 3 3 3 corresponding to the case n = 5 and indicates the presence of the functions (6.6.32) with s = 1, 2, 3 in the dynamical equations (6.6.46)–(6.6.54), etc. 6.7.

Case where the Moment of Nonconservative Forces Is Independent of the Angular Velocity

We choose the function rN as follows (the disk D n−1 is given by the equation x1N ≡ 0): ⎛ ⎞ 0 ⎜ x2N ⎟ ⎜ ⎟ rN = ⎜ .. ⎟ = R(α)iN , ⎝ . ⎠

(6.7.1)

xnN where iN = i v

π 2

 , β1 , . . . , βn−2

(6.7.2)

(see (6.4.2)). 329

In our case,

⎛

iN

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

0 cos β1 sin β1 cos β2 ... sin β1 . . . sin βn−3 cos βn−2 sin β1 . . . sin βn−2

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(6.7.3)

Thus, the equalities x2N = R(α) cos β1 ,

x3N = R(α) sin β1 cos β2 ,

xn−1,N = R(α) sin β1 . . . sin βn−3 cos βn−2 ,

...,

xnN = R(α) sin β1 . . . sin βn−2 ,

(6.7.4)

hold and show that, for the considered system, the moment of the nonconservative forces is independent of the tensor of angular velocity (it depends only on the angles α, β1 , . . . , βn−2 ). Thus, to construct the force field, we use the pair of dynamical functions R(α) and s(α); information on these functions has a qualitative nature. Similarly to the choice of the Chaplygin analytical functions (see [10, 122, 123]), we take the dynamical functions s and R as follows: R(α) = A sin α, 6.7.1.

s(α) = B cos α,

A, B > 0.

(6.7.5)

Reduced systems.

Theorem 6.7.1. The simultaneous equations (6.3.1), (6.4.3), and (6.5.9) under the conditions (6.3.6)–(6.3.8), (6.7.1), and (6.7.5) can be reduced to the dynamical system on the tangent bundle (6.1.5) of the (n − 1)-dimensional sphere (6.1.4). Indeed, if we introduce the dimensionless parameter and the differentiation by the formulas  AB , · = n0 v∞  , (6.7.6) b∗ = ln0 , n20 = (n − 2)I2 then the obtained equations have the following form: ξ  + b∗ ξ  cos ξ + sin ξ cos ξ 2 sin2 η1 . . . sin2 ηn−3 − η12 + η22 sin2 η1 + η32 sin2 η1 sin2 η2 + . . . + ηn−2

η1 + b∗ η1 cos ξ + ξ  η1

1 + cos2 ξ − cos ξ sin ξ

η2 + b∗ η2 cos ξ + ξ  η2

1 + cos2 ξ cos η1 + 2η1 η2 cos ξ sin ξ sin η1

η3 + b∗ η3 cos ξ + ξ  η3

1 + cos2 ξ cos η1 cos η2 + 2η1 η3 + 2η2 η3 cos ξ sin ξ sin η1 sin η2

! sin ξ cos ξ

= 0,

! 2 sin2 η2 . . . sin2 ηn−3 sin η1 cos η1 = 0, − η22 + η32 sin2 η2 + η42 sin2 η2 sin2 η3 + . . . + ηn−2

! 2 − η32 + η42 sin2 η3 + η52 sin2 η3 sin2 η4 + . . . + ηn−2 sin2 η3 . . . sin2 ηn−3 sin η2 cos η2 = 0,

! 2 − η42 + η52 sin2 η4 + η62 sin2 η4 sin2 η5 + . . . + ηn−2 sin2 η4 . . . sin2 ηn−3 sin η3 cos η3 = 0, (6.7.7) .....................................................................

330

1 + cos2 ξ cos η1  + 2η1 ηn−4 cos ξ sin ξ sin η1 ! cos ηn−5   2 2 2 + . . . + 2ηn−5 ηn−4 − ηn−3 + ηn−2 sin ηn−3 sin ηn−4 cos ηn−4 = 0, sin ηn−5

   ηn−4 + b∗ ηn−4 cos ξ + ξ  ηn−4

   + b∗ ηn−3 cos ξ + ξ  ηn−3 ηn−3

   + b∗ ηn−2 cos ξ + ξ  ηn−2 ηn−2

1 + cos2 ξ cos η1  + 2η1 ηn−3 cos ξ sin ξ sin η1 cos ηn−4   2 + . . . + 2ηn−4 ηn−3 − ηn−2 sin ηn−3 cos ηn−3 = 0, sin ηn−4 1 + cos2 ξ cos η1  + 2η1 ηn−2 cos ξ sin ξ sin η1   + . . . + 2ηn−3 ηn−2

cos ηn−3 = 0, sin ηn−3

b∗ > 0.

In particular, for n = 5 we have " # sin ξ = 0, ξ  + b∗ ξ  cos ξ + sin ξ cos ξ − η12 + η22 sin2 η1 + η32 sin2 η1 sin2 η2 cos ξ # 1 + cos2 ξ " 2 η1 + b∗ η1 cos ξ + ξ  η1 − η2 + η32 sin2 η2 sin η1 cos η1 = 0, cos ξ sin ξ 1 + cos2 ξ cos η1 + 2η1 η2 − η32 sin η2 cos η2 = 0, η2 + b∗ η2 cos ξ + ξ  η2 cos ξ sin ξ sin η1 1 + cos2 ξ cos η1 cos η2 + 2η1 η3 + 2η2 η3 = 0, b∗ > 0. η3 + b∗ η3 cos ξ + ξ  η3 cos ξ sin ξ sin η1 sin η2

(6.7.8)

After the transition from the variables z (see (6.5.7)) to the intermediate dimensionless variables w zk = n0 v∞ Zk ,

k = 1, . . . , n − 2,

zn−1 = n0 v∞ Zn−1 − n0 v∞ b∗ sin ξ,

(6.7.9)

the system (6.7.7) becomes equivalent to the system ξ  = Zn−1 − b∗ sin ξ,  2 = − sin ξ cos ξ + (Z12 + . . . + Zn−2 ) Zn−1

(6.7.10) cos ξ , sin ξ

cos ξ cos ξ cos η1 2 − (Z12 + . . . + Zn−3 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1  + Zn−3 Zn−2 = −Zn−3 Zn−1 Zn−3 sin ξ sin ξ sin η1 cos ξ 1 cos η2 2 + (Z12 + . . . + Zn−4 ) , sin ξ sin η1 sin η2 .......................................... n−2  cos ξ  cos ηs−1  s+1 (−1) Zn−s , Z1 = −Z1 sin ξ s=1 sin η1 . . . sin ηs−1  Zn−2 = −Zn−2 Zn−1

cos ξ , sin ξ cos ξ , η2 = Zn−3 sin ξ sin η1 .......................................... η1 = −Zn−2

(6.7.11) (6.7.12)

(6.7.13)

(6.7.14) (6.7.15) (6.7.16)

331

cos ξ , sin ξ sin η1 . . . sin ηn−4 cos ξ = (−1)n Z1 , sin ξ sin η1 . . . sin ηn−3

 ηn−3 = (−1)n+1 Z2

(6.7.17)

 ηn−2

(6.7.18)

on the tangent bundle T∗ Sn−1 {(Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) ∈ R2(n−1) : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π} (6.7.19) of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. We see that the independent subsystem (6.7.10)–(6.7.17) of the order 2(n − 1) − 1 (due to the fact that the variable ηn−2 is cyclic) can be substituted into the system (6.7.10)–(6.7.18) of the order 2(n − 1) and can be considered separately on its own (2n − 3)-dimensional manifold. In particular, for n = 5 we obtain the following eighth-order system: ξ  = Z4 − b∗ sin ξ,

(6.7.20)

cos ξ , sin ξ cos ξ cos η1 − (Z12 + Z22 ) , sin ξ sin η1 cos ξ cos η1 cos ξ 1 cos η2 + Z2 Z3 + Z12 , sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ cos η1 cos ξ 1 cos η2 + Z1 Z3 − Z1 Z2 , sin ξ sin η1 sin ξ sin η1 sin η2

Z4 = − sin ξ cos ξ + (Z12 + Z22 + Z32 ) cos ξ sin ξ cos ξ = −Z2 Z4 sin ξ cos ξ = −Z1 Z4 sin ξ cos ξ = −Z3 , sin ξ cos ξ = Z2 , sin ξ sin η1 cos ξ = −Z1 , sin ξ sin η1 sin η2

Z3 = −Z3 Z4 Z2 Z1 η1 η2 η3

(6.7.21) (6.7.22) (6.7.23) (6.7.24) (6.7.25) (6.7.26) (6.7.27)

on the tangent bundle T∗ S4 {(Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 , ) ∈ R8 : 0 ≤ ξ, η1 , η2 ≤ π,

η3 mod 2π}

(6.7.28)

of the four-dimensional sphere S4 {(ξ, η1 , η2 , η3 ) ∈ R4 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. We see that the independent seventh-order subsystem (6.7.20)–(6.7.26) (due to the fact that the variable η3 is cyclic) can be substituted into the eighth-order system (6.7.20)–(6.7.27) and can be considered separately on its own seven-dimensional manifold. 6.7.2. General remarks on the integrability of the system. In order to integrate the system (6.7.10)–(6.7.18) of the order 2(n − 1) completely, we must obtain, generally speaking, 2n − 3 independent first integrals (in particular, in order to integrate the eighth-order system (6.7.20)–(6.7.27) completely, we need, generally speaking, seven independent first integrals). But the systems considered have such symmetries that allow us to reduce a sufficient number of the first integrals down to n (in particular, down to five), in order to integrate the system. 332

6.7.2.1. System in the absence of a force field. Let us study the system (6.7.20)–(6.7.27) on the tangent bundle T∗ S4 {Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 } of the four-dimensional sphere S4 {ξ, η1 , η2 , η3 }; we also prove that this system is conservative. Furthermore, we assume that the function (6.6.15) identically vanishes (in particular, b∗ = 0, and also the coefficient sin ξ cos ξ in Eq. (6.7.21) is absent). The system studied has the form ξ  = Z4 , Z4 = (Z12 + Z22 + Z32 )

(6.7.29) cos ξ , sin ξ

(6.7.30)

cos ξ cos ξ cos η1 − (Z12 + Z22 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1 cos ξ 1 cos η2 = −Z2 Z4 + Z2 Z3 + Z12 , sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ cos ξ cos η1 cos ξ 1 cos η2 = −Z1 Z4 + Z1 Z3 − Z1 Z2 , sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ = −Z3 , sin ξ cos ξ = Z2 , sin ξ sin η1 cos ξ = −Z1 . sin ξ sin η1 sin η2

Z3 = −Z3 Z4

(6.7.31)

Z2

(6.7.32)

Z1 η1 η2 η3

(6.7.33) (6.7.34) (6.7.35) (6.7.36)

The system (6.7.29)–(6.7.36) describes the motion of a rigid body in the absence of an external force field. Theorem 6.7.2. The system (6.7.29)–(6.7.36) has five analytical independent first integrals as follows: $ (6.7.37) Φ1 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 + Z32 + Z42 = C1 = const, $ (6.7.38) Φ2 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 + Z32 sin ξ = C2 = const, $ (6.7.39) Φ3 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 sin ξ sin η1 = C3 = const, Φ4 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Z1 sin ξ sin η1 sin η2 = C4 = const,

(6.7.40)

Φ5 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = C5 = const .

(6.7.41)

These first integrals (6.7.37)–(6.7.40) show that if the external force field is absent, then four (in general, nonzero) components of the tensor of angular velocity of a five-dimensional rigid body are preserved, namely, ω4 ≡ ω40 = const,

ω7 ≡ ω70 = const,

ω9 ≡ ω90 = const,

0 ω10 ≡ ω10 = const .

(6.7.42)

In particular, the existence of the first integral (6.7.37) is explained by the equation Z12 + Z22 + Z32 + Z42 =

# 1 " 2 2 2 2 ω + ω + ω + ω ≡ C12 = const . 4 7 9 10 2 n20 v∞

(6.7.43)

The first integral (6.7.41) has the kinematic sense: it “attaches” the equation for η3 and can be found from the following quadrature: Z1 1 dη3 =− . dη2 Z2 sin η2

(6.7.44) 333

In this case, if we use the levels of the first integrals (6.7.39) and (6.7.40) and obtain the equality % Z1 =± Z2

C32 sin2 η2 − 1, C42

(6.7.45)

then the quadrature (6.7.44) has the form & η3 = ±

du '  C32 (1 − u2 ) − 1 − 2 C 4

C32 2 u C42

,

u = cos η2 .

(6.7.46)

The calculation of its quadrature implies the following form: η3 + C5 = ± arctg '

cos η2 C32 C42

2

,

C5 = const,

(6.7.47)

sin η2 − 1

which allows us to obtain the first integral (6.7.41). Transforming the last equality, we have the following invariant relation: tg2 (η3 + C5 ) =

C42 . (C32 − C42 ) tg2 η2 − C42

(6.7.48)

Now we reformulate Theorem 6.7.2 as follows. Theorem 6.7.3. The system (6.7.29)–(6.7.36) possesses five independent first integrals of the following form: Ψ1 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) =

Φ21 Z2 + Z2 + Z2 + Z2 = (1 2 2 2 3 2 4 = C1 = const, Φ2 Z1 + Z2 + Z3 sin ξ

Ψ2 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = C2 = const, ( Φ3 Z12 + Z22 = = C3 = const, Ψ3 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Φ4 Z1 sin η2 ( Φ2 Z 2 + Z22 + Z32 = ( 12 = C4 = const, Ψ4 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = Φ3 Z1 + Z22 sin η1 Ψ5 (Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) = C5 = const .

(6.7.49) (6.7.50) (6.7.51) (6.7.52) (6.7.53)

The first integral (6.7.53) also has the kinematic sense and “attaches” the equation for η3 , and the functions Ψ2 and Ψ5 can be selected equal to Φ2 and Φ5 , respectively. In the formulation of Theorem 6.7.3 (in contrast to Theorem 6.7.2), the smoothness characteristics of first integrals are absent. Namely, at points where the denominators (or both numerators and denominators simultaneously) of the first integrals (6.7.49)–(6.7.53) vanish, the integrals (considered as functions) have singularities. Furthermore, these functions are often discontinuous functions. By Theorem 6.7.3, the transformed set of first integrals (4.6.33)–(4.6.36) of the system (6.7.29)– (6.7.36) (i.e., the system in the absence of a force field) still remains a set of first integrals of the system studied. 334

To integrate the eighth-order system (6.7.29)–(6.7.36) completely, we need, in general, seven independent first integrals. However, after the change of variables w4 = −Z4 , $ w3 = Z32 + Z22 + Z12 , w2 =

Z2 , Z1

(6.7.54)

Z3 , w1 = − ( 2 Z2 + Z12 the system (6.7.29)–(6.7.36) splits as follows: ⎧  ξ = −w4 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  , w4 = −w32 sin ξ ⎪ ⎪ ⎪ cos ξ ⎪ ⎪ , ⎩w3 = w3 w4 sin ξ ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w2 cos η2 , 2 4 3 2 1 1 2 3 2 w2 sin η2 ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w1 cos η1 , 1 4 3 2 1 1 2 3 1 w1 sin η1 ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 1 4 3 2 1 1 2 3 1 η3 = d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ),

(6.7.55)

(6.7.56)

(6.7.57) (6.7.58)

where d1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = −Z3 (w4 , w3 , w2 , w1 )

cos ξ w1 w3 cos ξ = ∓( , sin ξ 1 + w12 sin ξ

cos ξ sin ξ sin η1 w2 w3 cos ξ ( = ±( , 2 2 1 + w1 1 + w2 sin ξ sin η1

d2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z2 (w4 , w3 , w2 , w1 )

(6.7.59)

cos ξ sin ξ sin η1 sin η2 w3 cos ξ ( = ∓( . 1 + w12 1 + w22 sin ξ sin η1 sin η2

d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = −Z1 (w4 , w3 , w2 , w1 )

In our case Zk = Zk (w4 , w3 , w2 , w1 ),

k = 1, 2, 3,

(6.7.60)

are functions by virtue of the change (6.7.54). The system (6.7.55)–(6.7.58) is studied on the tangent bundle T∗ S4 {(w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) ∈ R8 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}

(6.7.61)

of the four-dimensional sphere S4 {(ξ, η1 , η2 , η3 ) ∈ R4 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. We see that the independent third-order subsystem (6.7.55) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.7.56) 335

and (6.7.57) (after the change of independent variable) can be substituted into the eighth-order system (6.7.55)–(6.7.58), and also Eq. (6.7.58) for η3 is separated (due to the fact that the variable η3 is cyclic). Thus, to integrate the system (6.7.55)–(6.7.58) completely, it suffices to specify two independent first integrals of system (6.7.55), first integrals of the systems (6.7.56) and (6.7.57), and an additional first integral that “attaches” Eq. (6.7.58) (i.e., only five first integrals). Remark 6.7.1. We express the first integrals (6.7.49)–(6.7.53) in the variables w1 , w2 , w3 , and w4 by virtue of (6.7.54). We have Θ1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) =

w32 + w42 = C1 = const, w3 sin ξ

(6.7.62)

Θ2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = w3 sin ξ = C2 = const, ( 1 + w22 Θ3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = = C3 = const, sin η2 ( 1 + w12 = C4 = const, Θ4 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = sin η1

(6.7.63)

Θ5 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = C5 = const .

(6.7.66)

(6.7.64) (6.7.65)

Thus, two independent first integrals (6.7.62) and (6.7.63) are sufficient to integrate the system (6.7.55), the first integrals (6.7.64) and (6.7.65) are sufficient to integrate two independent first-order equations 1 + ws2 cos ηs dws = , s = 1, 2, (6.7.67) dηs ws sin ηs which is equivalent to the systems (6.7.56) and (6.7.57) after the change of independent variable, and, finally, the first integral (6.7.66) is sufficient “to attach” Eq. (6.7.58). We have proved the following theorem. Theorem 6.7.4. The eighth-order system (6.7.29)–(6.7.36) possesses a sufficient number (five) of independent first integrals. 6.7.2.2. System in the presence of a conservative force field. Now let us study the system (6.7.20)– (6.7.27) under the assumption b∗ = 0. In this case, we obtain a conservative system. Precisely, the coefficient sin ξ cos ξ in Eq. (6.7.21) (in contrast to the system (6.7.29)–(6.7.36)) characterizes the presence of the force field. The system studied has the form ξ  = Z4 ,

(6.7.68) cos ξ , sin ξ cos ξ cos η1 − (Z12 + Z22 ) , sin ξ sin η1 cos ξ cos η1 cos ξ 1 cos η2 + Z2 Z3 + Z12 , sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ cos η1 cos ξ 1 cos η2 + Z1 Z3 − Z1 Z2 , sin ξ sin η1 sin ξ sin η1 sin η2

Z4 = − sin ξ cos ξ + (Z12 + Z22 + Z32 ) cos ξ sin ξ cos ξ Z2 = −Z2 Z4 sin ξ cos ξ Z1 = −Z1 Z4 sin ξ cos ξ , η1 = −Z3 sin ξ Z3 = −Z3 Z4

336

(6.7.69) (6.7.70) (6.7.71) (6.7.72) (6.7.73)

cos ξ , (6.7.74) sin ξ sin η1 cos ξ η3 = −Z1 . (6.7.75) sin ξ sin η1 sin η2 Thus, the system (6.7.68)–(6.7.75) describes the motion of a rigid body in a conservative field of external forces. η2 = Z2

Theorem 6.7.5. The system (6.7.68)–(6.7.75) has the following five independent analytical first integrals: Φ1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 + Z32 + Z42 + sin2 ξ = C1 = const, $ Φ2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 + Z32 sin ξ = C2 = const, $ Φ3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z12 + Z22 sin ξ sin η1 = C3 = const,

(6.7.76)

Φ4 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z1 sin ξ sin η1 sin η2 = C4 = const,

(6.7.79)

Φ5 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = C5 = const .

(6.7.80)

(6.7.77) (6.7.78)

The first integral (6.7.76) is an integral of the total energy. The first integral (6.7.80) has the kinematic sense: it “attaches” the equation for η3 (see above). Now we reformulate Theorem 6.7.5 as follows. Theorem 6.7.6. The system (6.7.68)–(6.7.75) possesses five independent first integrals of the following form: Φ1 Z 2 + Z 2 + Z32 + Z42 + sin2 ξ = 1 ( 22 = C1 = const, Φ2 Z1 + Z22 + Z32 sin ξ Ψ2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = C2 = const, ( Φ3 Z12 + Z22 = = C3 = const, Ψ3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Φ4 Z1 sin η2 ( Φ2 Z 2 + Z22 + Z22 = ( 12 = C4 = const, Ψ4 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = 2 Φ3 Z1 + Z2 sin η1 Ψ1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) =

Ψ5 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = C5 = const .

(6.7.81) (6.7.82) (6.7.83) (6.7.84) (6.7.85)

The functions Ψ2 and Ψ5 can be selected equal to Φ2 and Φ5 , respectively. In the formulation of Theorem 6.7.6 (in contrast to Theorem 6.7.5), the smoothness characteristics of first integrals are absent. At points where the denominators (or both numerators and denominators simultaneously) of the first integrals (6.7.81)–(6.7.85) vanish, the integrals (considered as functions) have singularities. Furthermore, these functions are often discontinuous. By Theorem 6.7.6, the transformed set of the first integrals (6.7.81)–(6.7.85) of the system (6.7.68)– (6.7.75) (i.e., the system in the presence of a conservative force field) still remains as the set of the first integrals of the system studied. To integrate the eighth-order system (6.7.68)–(6.7.75) completely, we need, in general, seven independent first integrals. However, after the change of variables (6.7.54) the system (6.7.68)–(6.7.75) splits as follows: ⎧  ξ = −w4 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  , w4 = sin ξ cos ξ − w32 (6.7.86) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪ ⎪ , ⎩w3 = w3 w4 sin ξ 337

⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w2 cos η2 , 2 4 3 2 1 1 2 3 2 w2 sin η2 ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 2 4 3 2 1 1 2 3 2

(6.7.87)

⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w1 cos η1 , 1 4 3 2 1 1 2 3 1 w1 sin η1 ⎪  ⎩ η1 = d1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ),

(6.7.88)

η3 = d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ),

(6.7.89)

where the conditions (6.7.59) hold. The system (6.7.86)–(6.7.89) is studied on the tangent bundle (6.7.61) of the four-dimensional sphere 4 S {(ξ, η1 , η2 , η3 ) ∈ R4 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. We see that the independent third-order subsystem (6.7.86) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.7.87), (6.7.88) (after the change of independent variable) can be substituted into the eighth-order system (6.7.86)– (6.7.89), and also Eq. (6.7.89) for η3 is separated (due to the fact that the variable η3 is cyclic). Thus, to integrate completely the system (6.7.86)–(6.7.89), it suffices to specify two independent first integrals of the system (6.7.86), the first integrals of systems (6.7.87) and (6.7.88), and an additional first integral that “attaches” Eq. (6.7.89) (i.e., only five first integrals). Remark 6.7.2. We write the first integrals (6.7.81)–(6.7.85) in the variables w1 , w2 , w3 , and w4 by virtue of (6.7.54). We have w32 + w42 + sin2 ξ = C1 = const, w3 sin ξ Θ2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = w3 sin ξ = C2 = const, ( 1 + w22 = C3 = const, Θ3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = sin η2 ( 1 + w12 = C4 = const, Θ4 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = sin η1 Θ5 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = C5 = const .

Θ1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) =

(6.7.90) (6.7.91) (6.7.92) (6.7.93) (6.7.94)

Thus, two independent first integrals (6.7.90), (6.7.91) are sufficient to integrate the system (6.7.86), the first integrals (6.7.92), (6.7.93) are sufficient to integrate two independent first-order equations 1 + ws2 cos ηs dws = , dηs ws sin ηs

s = 1, 2,

(6.7.95)

which is equivalent to the systems (6.7.87) and (6.7.88)) after the change of independent variable, and, finally, the first integral (6.7.94) is sufficient “to attach” Eq. (6.7.89). We have proved the following theorem. Theorem 6.7.7. The eighth-order system (6.7.68)–(6.7.75) possesses a sufficient number (five) of independent first integrals. 6.7.3. Complete list of first integrals. Now we turn to integration of the desired eighth-order system (6.7.20)–(6.7.27) (without any simplifications, i.e., in the presence of all coefficients). 338

Similarly, to integrate the eighth-order system (6.7.20)–(6.7.27) completely, we need, in general, seven independent first integrals. However, after the change of variables (6.7.54), the system (6.7.20)– (6.7.27) splits as follows: ⎧  ⎪ξ = −w4 − b∗ sin ξ, ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  , w4 = sin ξ cos ξ − w32 (6.7.96) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪ ⎪ , ⎩w3 = w3 w4 sin ξ ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w2 cos η2 , 2 4 3 2 1 1 2 3 2 w2 sin η2 (6.7.97) ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w1 cos η1 , 1 4 3 2 1 1 2 3 1 w1 sin η1 (6.7.98) ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 1 4 3 2 1 1 2 3 1 η3 = d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ),

(6.7.99)

where the conditions (6.7.59) hold. The system (6.7.96)–(6.7.98) is studied on the tangent bundle (6.7.61) of the four-dimensional sphere S4 {(ξ, η1 , η2 , η3 ) ∈ R4 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. We see that the independent third-order subsystem (6.7.96) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.7.97) and (6.7.98) (after the change of independent variable) can be substituted into the eighth-order system (6.7.96)–(6.7.99), and also Eq. (6.7.99) for η3 is separated (due to the fact that the variable η3 is cyclic). Thus, to integrate the system (6.7.96)–(6.7.99) completely, it suffices to specify two independent first integrals of the system (6.7.96), the first integrals of the systems (6.7.97) and (6.7.98), and an additional first integral that “attaches” Eq. (6.7.99) (i.e., only five first integrals). First, we compare the third-order system (6.7.96) with the nonautonomous second-order system ⎧ sin ξ cos ξ − w32 cos ξ/ sin ξ dw4 ⎪ ⎪ = , ⎨ dξ −w4 − b∗ sin ξ (6.7.100) ⎪ w w cos ξ/ sin ξ dw ⎪ ⎩ 3 = 3 4 . dξ −w4 − b∗ sin ξ Using the substitution τ = sin ξ, we rewrite system (6.7.100) in the algebraic form: ⎧ 2 ⎪ ⎪ dw4 = τ − w3 /τ , ⎨ dτ −w4 − b∗ τ ⎪ w3 w4 /τ dw ⎪ ⎩ 3 = . dτ −w4 − b∗ τ Further, if we introduce the uniform variables by the formulas

(6.7.101)

w4 = u2 τ,

(6.7.102)

w3 = u1 τ,

we reduce system (6.7.101) to the following form: ⎧ 1 − u21 du2 ⎪ ⎪ + u2 = , ⎨τ dτ −u2 − b∗ ⎪ u1 u2 du ⎪ ⎩τ 1 + u1 = , dτ −u2 − b∗

(6.7.103)

339

which is equivalent to

⎧ 1 − u21 + u22 − bu2 du2 ⎪ ⎪ = , ⎨τ dτ −u2 − b∗ (6.7.104) ⎪ ⎪τ du1 = 2u1 u2 − bu1 . ⎩ dτ −u2 − b∗ We compare the second-order system (6.7.104) with the nonautonomous first-order equation du2 1 − u21 + u22 + b∗ u2 = , du1 2u1 u2 + b∗ u1

which can be easily reduced to the exact differential equation  2 u2 + u21 + b∗ u2 + 1 = 0. d u1

(6.7.105)

(6.7.106)

Therefore, Eq. (6.7.105) has the following first integral: u22 + u21 + b∗ u2 + 1 = C1 = const, u1

(6.7.107)

which in the old variables has the form Θ1 (w4 , w3 ; ξ) =

w42 + w32 + b∗ w4 sin ξ + sin2 ξ = C1 = const . w3 sin ξ

(6.7.108)

Remark 6.7.3. We consider system (6.7.96) with variable dissipation with zero mean (see [161, 168, 211]), which becomes conservative for b∗ = 0: ⎧  ξ = −w4 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  , w4 = sin ξ cos ξ − w32 (6.7.109) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪ ⎪ . ⎩w3 = w3 w4 sin ξ It has two analytical first integrals of the form w42 + w32 + sin2 ξ = C1∗ = const,

(6.7.110)

w3 sin ξ = C2∗ = const .

(6.7.111)

It is obvious that the ratio of the first integrals (6.7.110) and (6.7.111) is also a first integral of system (6.7.109). However, for b∗ = 0 both functions w42 + w32 + b∗ w4 sin ξ + sin2 ξ

(6.7.112)

and (6.7.111) are not first integrals of the system (6.7.96) but their ratio (i.e., the ratio of the functions (6.7.112) and (6.7.111)) is a first integral of system (6.7.96) for any b∗ . We find an obvious form of the additional first integral of the third-order system (6.7.96). For this, we first transform the invariant relation (6.7.107) for u1 = 0 as follows:   b∗ 2 C1 2 b2∗ + C12 − 1. (6.7.113) + u1 − = u2 + 2 2 4 We see that the parameters of the given invariant relation must satisfy the condition b2∗ + C12 − 4 ≥ 0,

(6.7.114)

and the phase space of system (6.7.96) is stratified into a family of surfaces defined by Eq. (6.7.113). 340

Thus, by virtue of relation (6.7.107), the first equation of system (6.7.104) has the form τ

2(1 + b∗ u2 + u22 ) − C1 U1 (C1 , u2 ) du2 = , dτ −u2 − b∗

(6.7.115)

where

$ 1 (6.7.116) {C1 ± C12 − 4(u22 + b∗ u2 + 1)}, 2 and the integration constant C1 is chosen from the condition (6.7.114). Therefore, the quadrature for the search of an additional first integral of system (6.7.96) has the form & & (−b∗ − u2 )du2 dτ ( = . (6.7.117) 2 τ 2(1 + b∗ u2 + u2 ) − C1 {C1 ± C12 − 4(u22 + b∗ u2 + 1)}/2 U1 (C1 , u2 ) =

Obviously, the left-hand side up to an additive constant is equal to ln | sin ξ|.

(6.7.118)

If b∗ = r1 , b21 = b2∗ + C12 − 4, (6.7.119) 2 then the right-hand side of Eq. (6.7.117) has the form & & d(b21 − 4r12 ) dr1 1 ( ( + b∗ − 2 2 2 2 2 2 4 (b1 − 4r1 ) ± C1 b1 − 4r1 (b1 − 4r1 ) ± C1 b21 − 4r12 (   b 1  b21 − 4r12  ∗ = − ln  ± 1 ± I1 , (6.7.120)  2  C1 2 u2 +

where

& I1 =

(

$

dr3 b21 − r32 (r3 ± C1 )

,

r3 =

b21 − 4r12 .

(6.7.121)

In the calculation of integral (6.7.121), the following three cases are possible. I. b∗ > 2. (   b2 − 4 + ( b2 − r 2  1 C   1 ∗ 1 3 ln  ±( I1 = − (  2 2  r3 ± C1 2 b∗ − 4 b∗ − 4  (   b2 − 4 − (b2 − r 2  C 1   1 ∗ 1 3 ln  ∓( + (  + const . (6.7.122) 2 2 r3 ± C1 2 b∗ − 4  b∗ − 4  II. b∗ < 2.

1 ±C1 r3 + b21 arcsin + const . I1 = ( b1 (r3 ± C1 ) 4 − b2∗

III. b∗ = 2.

(6.7.123)

(

I1 = ∓

b21 − r32 + const . C1 (r3 ± C1 )

(6.7.124)

When we return to the variable r1 =

w4 b∗ + , sin ξ 2

(6.7.125)

we obtain the final form for the value I1 : 341

I. b∗ > 2.

 (   b2 − 4 ± 2r  C   1 1 ( I1 = − ( ln  ( 2∗ ±  2 2 2 2 b∗ − 4  b1 − 4r1 ± C1 b∗ − 4   (   b2 − 4 ∓ 2r  1 C   1 ( 1  + const . (6.7.126) ln  ( 2∗ + ( ∓ 2 b2∗ − 4  b1 − 4r12 ± C1 b2∗ − 4  1

II. b∗ < 2.

( ±C1 b21 − 4r12 + b21 ( arcsin I1 = ( + const . 4 − b2∗ b1 ( b21 − 4r12 ± C1 ) 1

(6.7.127)

III. b∗ = 2.

2r1 ( + const . (6.7.128) 2 C1 ( b1 − 4r12 ± C1 ) Thus, we have found an additional first integral for the third-order system (6.7.96), i.e., we have a complete set of first integrals that are transcendental functions of the phase variables. I1 = ∓

Remark 6.7.4. In the expression of the found first integral, we must formally substitute the left-hand side of the first integral (6.7.107) instead of C1 . Then the obtained additional first integral has the following structure:  w4 w3 , = C2 = const . Θ2 (w4 , w3 ; ξ) = G sin ξ, sin ξ sin ξ

(6.7.129)

Thus, we have found two first integrals (6.7.108), (6.7.129) of the independent third-order system (6.7.96). To integrate it completely, it suffices to find one by one the first integral for the systems (6.7.97), (6.7.98), and an additional first integral that “attaches” Eq. (6.7.99). Indeed, the desired first integrals coincide with the first integrals (6.7.92)–(6.7.94), namely, ( 1 + w22 = C3 = const, (6.7.130) Θ3 (w2 ; η2 ) = sin η2 ( 1 + w12 = C4 = const, (6.7.131) Θ4 (w1 ; η1 ) = sin η1 C4 cos η2 Θ5 (w2 , w1 ; η1 , η2 , η3 ) = η3 ± arctg $ = C5 = const . (6.7.132) C32 sin2 η2 − C42 In this case, on the left-hand side of Eq. (6.7.132), we must substitute the first integrals (6.7.130) and (6.7.131) instead of C3 and C4 . Theorem 6.7.8. The eighth-order system (6.7.96)–(6.7.98) possesses a sufficient number (five) of the independent first integrals (6.7.108), (6.7.129), (6.7.130), (6.7.131), and (6.7.132). Therefore, in the considered case, the system of dynamical equations (6.7.20)–(6.7.27) has five first integrals expressed by the relations (6.7.108), (6.7.129), (6.7.130), (6.7.131), and (6.7.132), which are transcendental functions of the phase variables (in the sense of the complex analysis) and are expressed as a finite combination of elementary functions (in this case, we use the expressions (6.7.117)– (6.7.128)). Theorem 6.7.9. Three sets of relations (6.3.4), (6.4.4), and (6.5.10) under the conditions (6.3.6), (6.3.9), (6.3.10), (6.7.1), and (6.7.5) possess five the first integrals (a complete set), which are transcendental functions (in the sense of complex analysis) and are expressed as a finite combination of elementary functions. 342

6.7.4. Structure of equations on the tangent bundle of the finite-dimensional sphere. The study of complete system (6.7.20)–(6.7.27) on the tangent bundle T∗ S4 {Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 } of the four-dimensional sphere S4 {ξ, η1 , η2 , η3 } was started with the study of the simplified system (6.7.29)–(6.7.36) describing the dynamics under absence of any force field. Thus, the coefficients on the right-hand side of the system (6.7.29)–(6.7.36) have a geometrical sense only and are generated by the choice of coordinates Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 on the tangent bundle. The following question arises: How can vary the coefficients of corresponding systems under the inductive increasing of the dimension n − 1 of the sphere Sn−1 {ξ, η1 , . . . , ηn−2 }? In other words, of what kind of systems can the phase (geodesic) flows be described on the tangent bundle T∗ Sn−1 {Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 } of the (n − 1)-dimensional sphere Sn−1 {ξ, η1 , . . . , ηn−2 }, exactly in our chosen coordinates Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ? Although, we have considered in the explicit form the structure of the corresponding equations to the case n = 5 inclusively (in both this activity and the previous author activities), we begin from the case n = 2. This allows us to make the inductive step from n to n + 1 and “construct” similar systems of any higher order. Remark 6.7.5. Under construction of the systems on the tangent bundle T∗ Sn−1 {Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 } of the (n − 1)-dimensional sphere Sn−1 {ξ, η1 , . . . , ηn−2 }, we use the existence of the following tuple of analytical first integrals in our system: $ 2 = C1 = const, Φ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−1 $ 2 Φ2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−2 sin ξ = C2 = const, $ 2 sin ξ sin η1 = C3 = const, Φ3 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−3 (6.7.133) .............................................................................. $ Φn−2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + Z22 sin ξ sin η1 . . . sin ηn−4 = Cn−2 = const, Φn−1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z1 sin ξ sin η1 . . . sin ηn−3 = Cn−1 = const . These first integrals (6.7.133) state the fact that in the absence of an external force field, n − 1 components (in general, nonzero) of the tensor of angular velocity of an n-dimensional rigid body are preserved, namely, (6.7.134) ωr1 ≡ ωr01 = const, . . . , ωrn−1 ≡ ωr0n−1 = const . In particular, the existence of the first integral Φ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = C1 is explained by the equation 1 2 = 2 2 [ωr21 + . . . + ωr2n−1 ] ≡ C12 = const . (6.7.135) Z12 + . . . + Zn−1 n0 v∞ In this case, the first integrals (6.7.133) are functions on the components ωr1 , . . . , ωrn−1 . 6.7.4.1. Case n = 2. For n = 2, the following system defines a geodesic flow on the two-dimensional cylinder T∗ S1 {Z1 ; ξ}, which is the tangent bundle of the one-dimensional sphere S1 {ξ}: ξ  = Z1 , Z1 = 0.

(6.7.136)

In this case, by Remark 6.7.5, there exists the natural first integral Z1 = C1 = const .

(6.7.137) 343

The equation ξ  = Z1 is the kinematic relation and defines the coordinates ξ, Z1 in the phase space of system (6.7.136) (i.e., the tangent bundle T∗ S1 {Z1 ; ξ}). 6.7.4.2. Transition 2 → 3. Under the transition from n = 2 to n = 3, we change the notation Z1 → Z2 and introduce the new variable Z1 . Furthermore, in the desired system, the new terms appearing with increasing n will be underlined. Proposition 6.7.1. For n = 3, the following system defines the geodesic flow on the tangent bundle T∗ S2 {Z2 , Z1 ; ξ, η1 } of the two-dimensional sphere S2 {ξ, η1 }: ξ  = Z2 , cos ξ Z2 = Z12 , sin ξ cos ξ , Z1 = −Z1 Z2 sin ξ cos ξ η1 = −Z1 . sin ξ In this case, by Remark 6.7.5, there exist the first integrals

(6.7.138) (6.7.139) (6.7.140) (6.7.141)

Z12 + Z22 = C1 = const,

(6.7.142)

Z1 sin ξ = C2 = const .

(6.7.143)

Indeed, by virtue of (6.7.142), we have Z1 Z1 + Z2 Z2 = 0. Therefore, there exists a function N1 (ξ, η1 , Z1 , Z2 ) such that Z2 = −Z1 N1 (ξ, η1 , Z1 , Z2 ),

Z1 = Z2 N1 (ξ, η1 , Z1 , Z2 ),

and, by Eq. (6.7.143), the equality Z1 sin ξ + Z1 ξ  cos ξ = Z2 N1 (ξ, η1 , Z1 , Z2 ) sin ξ + Z1 Z2 cos ξ = 0 must hold (by virtue of the system (6.7.138)–(6.7.141)), whence N1 (ξ, η1 , Z1 , Z2 ) = −Z1

cos ξ , sin ξ

as required. Equations (6.7.138) and (6.7.141) are kinematic relations and define the coordinates ξ, η1 , Z1 , and Z2 in the phase space of the system (6.7.138)–(6.7.141) (i.e., the tangent bundle T∗ S2 {Z2 , Z1 ; ξ, η1 }). 6.7.4.3.

Transition 3 → 4. Under the transition from n = 3 to n = 4, we change the notation   Z3 Z2 → Z1 Z2

and introduce the new variable Z1 . Furthermore, in the desired system, the new terms appearing with increasing n will be underlined. Proposition 6.7.2. For n = 4, the following system defines a geodesic flow on the tangent bundle T∗ S3 {Z3 , Z2 , Z1 ; ξ, η1 , η2 } of the three-dimensional sphere S3 {ξ, η1 , η2 }: ξ  = Z3 , Z3 = (Z22 + Z12 ) 344

(6.7.144) cos ξ , sin ξ

(6.7.145)

cos ξ cos ξ cos η1 , − Z12 sin ξ sin ξ sin η1 cos ξ cos ξ cos η1 Z1 = −Z1 Z3 + Z1 Z2 , sin ξ sin ξ sin η1 cos ξ , η1 = −Z2 sin ξ cos ξ η2 = Z1 . sin ξ sin η1 Z2 = −Z2 Z3

(6.7.146) (6.7.147) (6.7.148) (6.7.149)

In this case, by Remark 6.7.5, there exist the first integrals Z12 + Z22 + Z32 = C1 = const, $ Z12 + Z22 sin ξ = C2 = const,

(6.7.150)

Z1 sin ξ sin η1 = C3 = const .

(6.7.152)

(6.7.151)

Indeed, by (6.7.150) and (6.7.151) and similarly to the proof of Proposition 6.7.1, we find the underlined term in Eq. (6.7.145), and we also conclude that Eqs. (6.7.146) and (6.7.147) take the following form: cos ξ cos ξ − Z12 N2 (ξ, η1 , η2 , Z1 , Z2 , Z3 ), Z2 = −Z2 Z3 sin ξ sin ξ (6.7.153) cos ξ cos ξ  + Z1 Z2 N2 (ξ, η1 , η2 , Z1 , Z2 , Z3 ). Z1 = −Z1 Z3 sin ξ sin ξ Further, by Eqs. (6.7.152) and (6.7.144)–(6.7.149), the relations Z1 sin ξ sin η1 + Z1 ξ  cos ξ sin η1 + Z1 η1 sin ξ cos η1

! = Z1 Z2 cos ξ − cos η1 + N2 (ξ, η1 , η2 , Z1 , Z2 , Z3 ) sin η1 = 0,

are valid (by virtue of the system (6.7.144)–(6.7.149)) since N2 (ξ, η1 , η2 , Z1 , Z2 , Z3 ) =

cos η1 , sin η1

as required. Equations (6.7.144), (6.7.148), and (6.7.149) are kinematic relations; they define the coordinates ξ, η1 , η2 , Z1 , Z2 , and Z3 in the phase space of system (6.7.144)–(6.7.149) (i.e., the tangent bundle T∗ S3 {Z3 , Z2 , Z1 ; ξ, η1 , η2 }). Transition 4 → 5. Under the transition from n = 4 to n = 5, we change the notation as ⎞ ⎛ ⎞ ⎛ Z4 Z3 ⎝ Z2 ⎠ → ⎝ Z3 ⎠ Z1 Z2 and introduce the new variable Z1 . Furthermore, in the desired system, the new terms appearing with increasing n will be underlined. 6.7.4.4. follows:

Proposition 6.7.3. For n = 5, the following system defines a geodesic flow on the tangent bundle T∗ S4 {Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 } of the four-dimensional sphere S4 {ξ, η1 , η2 , η3 }: ξ  = Z4 , Z4 = (Z32 + Z22 + Z12 )

(6.7.154) cos ξ , sin ξ

(6.7.155) 345

cos ξ cos η1 cos ξ , − (Z22 + Z12 ) sin ξ sin ξ sin η1 cos ξ cos ξ cos η1 cos ξ 1 cos η2 Z2 = −Z2 Z4 + Z2 Z3 + Z12 , sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ cos ξ cos η1 cos ξ 1 cos η2 − Z1 Z2 , + Z1 Z3 Z1 = −Z1 Z4 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ η1 = −Z3 , sin ξ cos ξ , η2 = Z2 sin ξ sin η1 cos ξ η3 = −Z1 . sin ξ sin η1 sin η2 In this case, by Remark 6.7.5, there exist the first integrals Z3 = −Z3 Z4

(6.7.156) (6.7.157) (6.7.158) (6.7.159) (6.7.160) (6.7.161)

Z12 + Z22 + Z32 + Z42 = C1 = const, $ Z12 + Z22 + Z32 sin ξ = C2 = const, $ Z12 + Z22 sin ξ sin η1 = C3 = const,

(6.7.162)

Z1 sin ξ sin η1 sin η2 = C4 = const .

(6.7.165)

(6.7.163) (6.7.164)

Indeed, by virtue of (6.7.162)–(6.7.164) and similarly to the proofs of Propositions 6.7.1 and 6.7.2, we find the underlined terms in Eqs. (6.7.155) and (6.7.156) and also conclude that Eqs. (6.7.157) and (6.7.158) take the following forms: cos ξ cos ξ cos η1 cos ξ + Z2 Z3 N3 (ξ, η1 , η2 , η3 , Z1 , Z2 , Z3 , Z4 ), + Z12 Z2 = −Z2 Z4 sin ξ sin ξ sin η1 sin ξ (6.7.166) cos ξ cos ξ cos η1 cos ξ  + Z1 Z3 − Z1 Z2 N3 (ξ, η1 , η2 , η3 , Z1 , Z2 , Z3 , Z4 ). Z1 = −Z1 Z4 sin ξ sin ξ sin η1 sin ξ Further, by Eq. (6.7.165), the equality Z1 sin ξ sin η1 sin η2 + Z1 ξ  cos ξ sin η1 sin η2 + Z1 η1 sin ξ cos η1 sin η2 + Z1 η2 sin ξ sin η1 cos η2 ! = Z1 Z2 cos ξ − N3 (ξ, η1 , η2 , η3 , Z1 , Z2 , Z3 , Z4 ) sin η1 sin η2 + cos η2 = 0, must hold (by virtue of the system (6.7.154)–(6.7.161)), since N3 (ξ, η1 , η2 , η3 , Z1 , Z2 , Z3 , Z4 ) =

1 cos η2 , sin η1 sin η2

as required. Equations (6.7.154) and (6.7.159)–(6.7.161) are kinematic relations; they define the coordinates ξ, η1 , η2 , η3 , Z1 , Z2 , Z3 , and Z4 in the phase space of the system (6.7.154)–(6.7.161) (i.e., the tangent bundle T∗ S4 {Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 }). Transition 5 → 6. Under the transition from n = 5 to n = 6, we change the notation as ⎞ ⎛ ⎞ ⎛ Z5 Z4 ⎜ ⎟ ⎜ Z3 ⎟ ⎟ → ⎜ Z4 ⎟ ⎜ ⎝ Z3 ⎠ ⎝ Z2 ⎠ Z1 Z2 and introduce the new variable Z1 . Furthermore, in the desired system, the new terms appearing with increasing n will be underlined. 6.7.4.5. follows:

346

Proposition 6.7.4. For n = 6, the following system defines the geodesic flow on the tangent bundle T∗ S5 {Z5 , Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 , η4 } of the five-dimensional sphere S5 {ξ, η1 , η2 , η3 , η4 }: ξ  = Z5 ,

(6.7.167)

Z5 = (Z42 + Z32 + Z22 + Z12 )

cos ξ , sin ξ

(6.7.168)

cos ξ cos ξ cos η1 − (Z32 + Z22 + Z12 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z3 Z4 + (Z22 + Z12 ) , Z3 = −Z3 Z5 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z2 Z4 Z2 = −Z2 Z5 − Z2 Z3 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ 1 1 cos η3 −Z12 , sin ξ sin η1 sin η2 sin η3 cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z1 Z4 − Z1 Z3 Z1 = −Z1 Z5 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ 1 1 cos η3 +Z1 Z2 , sin ξ sin η1 sin η2 sin η3 cos ξ ,7 7η1 = −Z4 sin ξ cos ξ , η2 = Z3 sin ξ sin η1 cos ξ η3 = −Z2 , sin ξ sin η1 sin η2 cos ξ . η4 = Z1 sin ξ sin η1 sin η2 sin η3 Z4 = −Z4 Z5

(6.7.169) (6.7.170)

(6.7.171)

(6.7.172) (6.7.173) (6.7.174) (6.7.175) (6.7.176)

In this case, by Remark 6.7.5, there exist the first integrals Z12 + Z22 + Z32 + Z42 + Z52 $ Z12 + Z22 + Z32 + Z42 sin ξ $ Z12 + Z22 + Z32 sin ξ sin η1 $ Z12 + Z22 sin ξ sin η1 sin η2

= C1 = const,

(6.7.177)

= C2 = const,

(6.7.178)

= C3 = const,

(6.7.179)

= C4 = const,

(6.7.180)

Z1 sin ξ sin η1 sin η2 sin η3 = C5 = const .

(6.7.181)

Indeed, by virtue of (6.7.177)–(6.7.180) and similarly to the proofs of Propositions 6.7.1–6.7.3, we find the underlined terms in Eqs. (6.7.168)–(6.7.170) and also conclude that Eqs. (6.7.171) and (6.7.172) take the following forms: Z2 = −Z2 Z5

cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z2 Z4 − Z2 Z3 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ N4 (ξ, η1 , η2 , η3 , η4 , Z1 , Z2 , Z3 , Z4 , Z5 ), (6.7.182) − Z12 sin ξ 347

Z1 = −Z1 Z5

cos ξ cos ξ cos η1 cos ξ 1 cos η2 − Z1 Z3 + Z1 Z4 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ + Z1 Z2 N4 (ξ, η1 , η2 , η3 , η4 , Z1 , Z2 , Z3 , Z4 , Z5 ). sin ξ

Further, by Eq. (6.7.181), the equality Z1 sin ξ sin η1 sin η2 sin η3 + Z1 ξ  cos ξ sin η1 sin η2 sin η3 + Z1 η1 sin ξ cos η1 sin η2 sin η3 + Z1 η2 sin ξ sin η1 cos η2 sin η3 + Z1 η3 sin ξ sin η1 sin η2 cos η3

! = Z1 Z2 cos ξ − cos η3 + N4 (ξ, η1 , η2 , η3 , η4 , Z1 , Z2 , Z3 , Z4 , Z5 ) sin η1 sin η2 sin η3 = 0 must hold (by virtue of system (6.7.167)–(6.7.176)), since N4 (ξ, η1 , η2 , η3 , η4 , Z1 , Z2 , Z3 , Z4 , Z5 ) =

1 cos η3 1 , sin η1 sin η2 sin η3

as required. Equations (6.7.167) and (6.7.173)–(6.7.176) are kinematic relations; they define the coordinates ξ, η1 , η2 , η3 , η4 , Z1 , Z2 , Z3 , Z4 , and Z5 in the phase space of the system (6.7.167)–(6.7.176) (i.e., the tangent bundle T∗ S5 {Z5 , Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 , η4 }). 6.7.4.6.

Transition n → n+1. Under the inductive transition from n to n+1, we change the notation ⎞ ⎛ ⎞ ⎛ Zn Zn−1 ⎜ ⎟ ⎜ Zn−2 ⎟ ⎟ → ⎜ Zn−1 ⎟ ⎜ ⎝ ... ⎠ ⎝ ... ⎠ Z1 Z2

and introduce the new variable Z1 . Furthermore, in the desired system, the new terms appearing with increasing n will be underlined. Proposition 6.7.5. For n > 2, the following system defines the geodesic flow on the tangent bundle T∗ Sn {Zn , Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−1 } of the n-dimensional sphere Sn {ξ, η1 , . . . , ηn−1 }: ξ  = Zn , 2 + . . . + Z22 + Z12 ) Zn = (Zn−1

(6.7.183) cos ξ , sin ξ

cos ξ cos η1 cos ξ 2 − (Zn−2 + . . . + Z22 + Z12 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1  + Zn−2 Zn−1 = −Zn−2 Zn Zn−2 sin ξ sin ξ sin η1 cos ξ 1 cos η2 2 + (Zn−3 + . . . + Z22 + Z12 ) , sin ξ sin η1 sin η2 ................................................................................. cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z2 Zn−1 − Z2 Zn−2 Z2 = −Z2 Zn sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ 1 1 cos ηn−3 + . . . + (−1)n Z2 Z3 ... sin ξ sin η1 sin ηn−4 sin ηn−3 1 cos ξ 1 cos ηn−2 + (−1)n Z12 ... , sin ξ sin η1 sin ηn−3 sin ηn−2  Zn−1 = −Zn−1 Zn

348

(6.7.184) (6.7.185)

(6.7.186)

(6.7.187)

Z1 = −Z1 Zn

cos ξ cos ξ cos η1 cos ξ 1 cos η2 −Z1 Zn−2 + ...+ + Z1 Zn−1 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ 1 cos ηn−3 1 +(−1)n Z1 Z3 ... sin ξ sin η1 sin ηn−4 sin ηn−3 +(−1)n+1 Z1 Z2

cos ξ 1 1 cos ηn−2 ... , sin ξ sin η1 sin ηn−3 sin ηn−2

cos ξ , sin ξ cos ξ η2 = Zn−2 , sin ξ sin η1 .............................. cos ξ  = (−1)n Z2 , ηn−2 sin ξ sin η1 . . . sin ηn−3 cos ξ  ηn−1 = (−1)n+1 Z1 . sin ξ sin η1 . . . sin ηn−2 η1 = −Zn−1

(6.7.188)

(6.7.189) (6.7.190)

(6.7.191) (6.7.192)

In this case, by Remark 6.7.5, there exist the first integrals Z12 + . . . + Zn2 = C1 = const,

(6.7.193)

2 Z12 + . . . + Zn−1 sin ξ = C2 = const,

(6.7.194)

2 Z12 + . . . + Zn−2 sin ξ sin η1 = C3 = const,

(6.7.195)

$ $ $

.............................. Z12 + Z22 sin ξ sin η1 . . . sin ηn−3 = Cn−1 = const, Z1 sin ξ sin η1 . . . sin ηn−2 = Cn = const .

(6.7.196) (6.7.197)

Indeed, by virtue of (6.7.193)–(6.7.196) and similarly to the proofs of Propositions 6.7.1–6.7.4, we find the underlined terms in all equations up to Eqs. (6.7.187) and (6.7.188), and we also conclude that Eqs. (6.7.187) and (6.7.188) take the following forms: Z2 = −Z2 Zn

cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z2 Zn−1 − Z2 Zn−2 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 1 1 cos ξ cos ηn−3 + . . . + (−1)n Z2 Z3 ... sin ξ sin η1 sin ηn−4 sin ηn−3 cos ξ Nn−1 (ξ, η1 , . . . , ηn−1 , Z1 , . . . , Zn ), (6.7.198) + (−1)n Z12 sin ξ

Z1 = −Z1 Zn

cos ξ cos ξ cos η1 cos ξ 1 cos η2 + Z1 Zn−1 − Z1 Zn−2 sin ξ sin ξ sin η1 sin ξ sin η1 sin η2 cos ξ 1 1 cos ηn−3 + . . . + (−1)n Z1 Z3 ... sin ξ sin η1 sin ηn−4 sin ηn−3 cos ξ Nn−1 (ξ, η1 , . . . , ηn−1 , Z1 , . . . , Zn ). + (−1)n+1 Z1 Z2 sin ξ 349

Further, by Eq. (6.7.197), the equality Z1 sin ξ sin η1 . . . sin ηn−2 + Z1 ξ  cos ξ sin η1 . . . sin ηn−2 + Z1 η1 sin ξ cos η1 sin η2 . . . sin ηn−2 + . . .   + Z1 ηn−3 sin ξ sin η1 . . . sin ηn−4 cos ηn−3 sin ηn−2 + Z1 ηn−2 sin ξ sin η1 . . . sin ηn−3 cos ηn−2

= (−1)n Z1 Z2 cos ξ[cos ηn−2 − Nn−1 (ξ, η1 , . . . , ηn−1 , Z1 , . . . , Zn ) sin η1 . . . sin ηn−2 ] = 0

must hold (by virtue of system (6.7.183)–(6.7.192)), whence 1 1 1 cos ηn−2 ... , Nn−1 (ξ, η1 , . . . , ηn−1 , Z1 , . . . , Zn ) = sin η1 sin η2 sin ηn−3 sin ηn−2 as required. Equations (6.7.183) and (6.7.189)–(6.7.192) are kinematic relations; they define the coordinates ξ, η1 , . . ., ηn−1 , Z1 , . . ., Zn in the phase space of the system (6.7.183)–(6.7.192) (i.e., the tangent bundle T∗ Sn {Zn , . . . , Z1 ; ξ, η1 , . . . , ηn−1 }). 6.7.5. General remarks on the integrability of the system for any finite n. As was already mentioned, in order to integrate the system (6.7.10)–(6.7.18) of the order 2(n − 1) completely, we must obtain, generally speaking, 2n − 3 independent first integrals. However, the system considered has certain symmetries that allow us to reduce its order and a sufficient number of first integrals down to n. 6.7.5.1. System in the absence of a force field. Let us study the system (6.7.10)–(6.7.18) on the tangent bundle T∗ Sn−1 {Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 } of the (n − 1)-dimensional sphere Sn−1 {ξ, η1 , . . . , ηn−2 }. At the same time, we prove that this system is conservative. Furthermore, we assume that the function (6.6.15) identically vanishes (in particular, b∗ = 0, and also the coefficient sin ξ cos ξ in Eq. (6.7.11) is absent). The system studied has the form ξ  = Zn−1 ,  2 = (Z12 + . . . + Zn−2 ) Zn−1

(6.7.199) cos ξ , sin ξ

cos ξ cos ξ cos η1 2 − (Z12 + . . . + Zn−3 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1  = −Zn−3 Zn−1 + Zn−3 Zn−2 Zn−3 sin ξ sin ξ sin η1 cos ξ 1 cos η2 2 + (Z12 + . . . + Zn−4 ) , sin ξ sin η1 sin η2 ...................................................... n−2   cos ξ cos η s−1 (−1)s+1 Zn−s , Z1 = −Z1 sin ξ s=1 sin η1 . . . sin ηs−1  Zn−2 = −Zn−2 Zn−1

cos ξ , sin ξ cos ξ , η2 = Zn−3 sin ξ sin η1 ...................................................... cos ξ  = (−1)n+1 Z2 , ηn−3 sin ξ sin η1 . . . sin ηn−4 cos ξ  = (−1)n Z1 . ηn−2 sin ξ sin η1 . . . sin ηn−3 η1 = −Zn−2

350

(6.7.200) (6.7.201)

(6.7.202)

(6.7.203) (6.7.204) (6.7.205)

(6.7.206) (6.7.207)

The system (6.7.199)–(6.7.207) describes the motion of a rigid body in the absence of an external force field. Theorem 6.7.10. The system (6.7.199)–(6.7.207) has n analytical independent first integrals as follows: $ 2 = C1 = const, (6.7.208) Φ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−1 $ 2 Φ2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−2 sin ξ = C2 = const, (6.7.209) $ 2 Φ3 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−3 sin ξ sin η1 = C3 = const, (6.7.210) ............................................................ $ Φn−2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + Z22 sin ξ sin η1 . . . sin ηn−4 = Cn−2 = const,

(6.7.211)

Φn−1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z1 sin ξ sin η1 . . . sin ηn−3 = Cn−1 = const,

(6.7.212)

Φn (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

(6.7.213)

These first integrals (6.7.208)–(6.7.212) state that in the absence of an external force field, n − 1 components (in general, nonzero) of the tensor of angular velocity of an n-dimensional rigid body are preserved, namely, (6.7.214) ωr1 ≡ ωr01 = const, . . . , ωrn−1 ≡ ωr0n−1 = const . In particular, the existence of the first integral (6.7.208) is explained by the equation 2 = Z12 + . . . + Zn−1

1 [ωr21 2 2 n0 v∞

+ . . . + ωr2n−1 ] ≡ C12 = const .

(6.7.215)

The first integral (6.7.213) has the kinematic sense: it “attaches” the equation for ηn−2 and can be found from the following quadrature: Z1 1 dηn−2 =− . dηn−3 Z2 sin ηn−3

(6.7.216)

In this case, if we use the levels of the first integrals (6.7.211) and (6.7.212) and obtain the equality % 2 Cn−2 Z1 =± sin2 ηn−3 − 1, (6.7.217) 2 Z2 Cn−1 then the quadrature (6.7.216) has the form & du ' ηn−2 = ±  2 Cn−2 (1 − u2 ) − 1 − C2 n−1

2 Cn−2 u2 2 Cn−1

The calculation of its quadrature implies the following form: cos ηn−3 , ηn−2 + Cn = ± arctg ' 2 Cn−2 2 sin ηn−3 − 1 C2

,

u = cos ηn−3 .

Cn = const,

(6.7.218)

(6.7.219)

n−1

which allows us to obtain the first integral (6.7.213). Transforming the last equality, we have the following invariant relation: tg2 (ηn−2 + Cn ) =

2 Cn−1 . 2 2 ) tg2 η 2 (Cn−2 − Cn−1 n−3 − Cn−1

(6.7.220)

Now we reformulate Theorem 6.7.10 as follows. 351

Theorem 6.7.11. The system (6.7.199)–(6.7.207) possesses n independent first integrals of the following form: Ψ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) =

2 Z 2 + . . . + Zn−1 Φ21 =$ 1 = C1 = const, Φ2 2 2 Z1 + . . . + Zn−2 sin ξ

Ψ2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = C2 = const, ( Φn−2 Z12 + Z22 = = C3 = const, Ψ3 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Φn−1 Z1 sin ηn−3 ............................................................ $ 2 Z12 + . . . + Zn−3 Φ3  =$ = Cn−2 = const, Ψn−2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Φ4 Z 2 + . . . + Z 2 sin η 1

$

n−4

(6.7.222) (6.7.223)

(6.7.224)

2

2 Z12 + . . . + Zn−2 Φ2  $ Ψn−1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = = = Cn−1 = const, Φ3 2 Z12 + . . . + Zn−3 sin η1

Ψn (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

(6.7.221)

(6.7.225) (6.7.226)

The first integral (6.7.226) has also the kinematic sense and “attaches” the equation for ηn−2 , and the functions Ψ2 and Ψn can be selected equal to Φ2 and Φn , respectively. In the formulation of Theorem 6.7.11 (in contrast to Theorem 6.7.10), the smoothness characteristics of first integrals are absent. Precisely, at points where the denominators (or the numerators and denominators simultaneously) of the first integrals (6.7.221)–(6.7.226) vanish, the integrals (considered as functions) have singularities. Furthermore, these functions are often discontinuous. By Theorem 6.7.11, the transformed set of the first integrals (6.7.221)–(6.7.226) of the system (6.7.199)–(6.7.207) (i.e., the system in the absence of a force field) still remains the set of first integrals of the system studied. To integrate the system (6.7.199)–(6.7.207) of the order 2(n − 1) completely, we need, in general, 2n − 3 independent first integrals. However, after the following change of variables ⎞ ⎛ ⎞ ⎛ wn−1 Zn−1 ⎜ wn−2 ⎟ ⎜ Zn−2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ... ⎟ → ⎜ ... ⎟, ⎟ ⎜ ⎟ ⎜ ⎝ w2 ⎠ ⎝ Z2 ⎠ Z1 w1 $ Z2 Z3 2 , wn−3 = , wn−4 = − ( 2 , ..., wn−1 = −Zn−1 , wn−2 = Z12 + . . . + Zn−2 Z1 Z1 + Z22 (6.7.227) Zn−3 Zn−2 , w1 = − $ , w2 = − $ 2 2 Z12 + . . . + Zn−4 Z12 + . . . + Zn−3 the system (6.7.199)–(6.7.207) splits as follows: ⎧  ξ = −wn−1 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 , wn−1 = −wn−2 sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ , = wn−2 wn−1 ⎩wn−2 sin ξ 352

(6.7.228)

⎧ ⎪ 1 + ws2 cos ηs ⎨w = d (w , . . . , w ; ξ, η , . . . , η ) , s n−1 1 1 n−2 s ws sin ηs ⎪ ⎩η  = d (w s = 1, . . . , n − 3, s n−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ), s  ηn−2 = dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ),

(6.7.229) (6.7.230)

where cos ξ , sin ξ cos ξ d2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = Zn−3 (wn−1 , . . . , w1 ) , sin ξ sin η1 ............................................................

d1 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = −Zn−2 (wn−1 , . . . , w1 )

dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = (−1)n Z1 (wn−1 , . . . , w1 )

(6.7.231)

cos ξ . sin ξ sin η1 . . . sin ηn−3

In our case Zk = Zk (wn−1 , . . . , w1 ), k = 1, . . . , n − 2, are functions by virtue of change (6.7.227). The system (6.7.228)–(6.7.230) is studied on the tangent bundle

(6.7.232)

T∗ Sn−1 {(wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) ∈ R2(n−1) : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π} (6.7.233) 0 ≤ ξ, η1 , . . . , ηn−3 ≤ of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : π, ηn−2 mod 2π}. We see that the independent third-order subsystem (6.7.228) (which can be considered separately on its own three-dimensional manifold) and n−3 independent second-order subsystems (6.7.229) (after the change of independent variable) can be substituted into the system (6.7.228)–(6.7.230) of the order 3 + 2(n − 3) + 1 = 2(n − 1), and also Eq. (6.7.230) for ηn−2 is separated (due to the fact that the variable ηn−2 is cyclic). Thus, to integrate the system (6.7.228)–(6.7.230) completely, it suffices to specify two independent first integrals of system (6.7.228), first integrals of the systems (6.7.229) (their total number is n − 3), and an additional first integral that “attaches” Eq. (6.7.230) (i.e., only n first integrals). Remark 6.7.6. Rewriting the first integrals (6.7.221)–(6.7.226) in the variables w1 , . . . , wn−1 by virtue of (6.7.227), we obtain 2 2 wn−2 + wn−1 = C1 = const, wn−2 sin ξ Θ2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = wn−2 sin ξ = C2 = const, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, Θs+2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = sin ηs Θn (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

Θ1 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) =

(6.7.234) (6.7.235) (6.7.236) (6.7.237)

Thus, two independent first integrals (6.7.234), (6.7.235) are sufficient to integrate the system (6.7.228), the n − 3 first integrals (6.7.236) are sufficient to integrate the independent first-order equations 1 + ws2 cos ηs dws = , s = 1, . . . , n − 3, (6.7.238) dηs ws sin ηs which are equivalent to the systems (6.7.229) after the change of independent variable, and, finally, the first integral (6.7.237) is sufficient “to attach” Eq. (6.7.230). We have proved the following theorem. Theorem 6.7.12. The system (6.7.199)–(6.7.207) of the order 2(n − 1) possesses a sufficient number (namely, n) of independent first integrals. 353

6.7.5.2. System in the presence of a conservative force field. Now we examine the system (6.7.10)– (6.7.18) under the assumption b∗ = 0. In this case, we obtain a conservative system. Namely, the coefficient sin ξ cos ξ in Eq. (6.7.11) (in contrast to the system (6.7.199)–(6.7.207)) characterizes the presence of the force field. The system studied has the form ξ  = Zn−1 ,

(6.7.239)

 2 = − sin ξ cos ξ + (Z12 + . . . + Zn−2 ) Zn−1

cos ξ , sin ξ

cos ξ cos ξ cos η1 2 − (Z12 + . . . + Zn−3 ) , sin ξ sin ξ sin η1 cos ξ cos ξ cos η1  = −Zn−3 Zn−1 Zn−3 + Zn−3 Zn−2 sin ξ sin ξ sin η1 cos ξ 1 cos η2 2 + (Z12 + . . . + Zn−4 ) , sin ξ sin η1 sin η2 ................................................... n−2   cos ξ cos η s−1 (−1)s+1 Zn−s , Z1 = −Z1 sin ξ sin η1 . . . sin ηs−1  Zn−2 = −Zn−2 Zn−1

(6.7.240) (6.7.241)

(6.7.242)

(6.7.243)

s=1

cos ξ , sin ξ cos ξ , η2 = Zn−3 sin ξ sin η1 ................................................... cos ξ  = (−1)n+1 Z2 , ηn−3 sin ξ sin η1 . . . sin ηn−4 cos ξ  ηn−2 = (−1)n Z1 . sin ξ sin η1 . . . sin ηn−3 η1 = −Zn−2

(6.7.244) (6.7.245)

(6.7.246) (6.7.247)

Thus, the system (6.7.239)–(6.7.247) describes the motion of a rigid body in a conservative field of external forces. Theorem 6.7.13. The system (6.7.239)–(6.7.247) has the following n independent analytical first integrals: 2 + sin2 ξ = C1 = const, Φ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−1 $ 2 sin ξ = C2 = const, Φ2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−2 $ 2 sin ξ sin η1 = C3 = const, Φ3 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + . . . + Zn−3

(6.7.248) (6.7.249) (6.7.250)

............................................................ $ Φn−2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z12 + Z22 sin ξ sin η1 . . . sin ηn−4 = Cn−2 = const,

(6.7.251)

Φn−1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Z1 sin ξ sin η1 . . . sin ηn−3 = Cn−1 = const,

(6.7.252)

Φn (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

(6.7.253)

The first integral (6.7.248) is the total energy integral. The first integral (6.7.253) has the kinematic sense: it “attaches” the equation for βn−2 (see above). Now we reformulate Theorem 6.7.13 as follows. 354

Theorem 6.7.14. The system (6.7.239)–(6.7.247) possesses n independent first integrals of the following form: Ψ1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) =

2 Z12 + . . . + Zn−1 + sin2 ξ Φ1 = $ = C1 = const, Φ2 2 2 Z1 + . . . + Zn−2 sin ξ

Ψ2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = C2 = const, ( Φn−2 Z12 + Z22 = = C3 = const, Ψ3 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Φn−1 Z1 sin ηn−3 ............................................................ $ 2 Z12 + . . . + Zn−3 Φ3  =$ = Cn−2 = const, Ψn−2 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Φ4 2 2 Z + ... + Z sin η 1

$

n−4

(6.7.255) (6.7.256)

(6.7.257)

2

2 Z12 + . . . + Zn−2 Φ2  Ψn−1 (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = =$ = Cn−1 = const, Φ3 2 2 Z1 + . . . + Zn−3 sin η1

Ψn (Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

(6.7.254)

(6.7.258) (6.7.259)

The functions Ψ2 and Ψn can be selected equal to Φ2 and Φn , respectively. In the formulation of Theorem 6.7.14 (in contrast to Theorem 6.7.13), the smoothness characteristics of first integrals are absent. Precisely, at points where the denominators (or both numerators and denominators simultaneously) of the first integrals (6.7.254)–(6.7.259) vanish, the integrals (considered as functions) have singularities. Furthermore, these functions are often discontinuous. By Theorem 6.7.14, the transformed set of the first integrals (6.7.254)–(6.7.259) of the system (6.7.239)–(6.7.247) (i.e., the system in the presence of a conservative force field) still remains a set of first integrals of the system studied. To integrate the system (6.7.239)–(6.7.247) of the order 2(n − 1) completely, we need, in general, 2n − 3 independent first integrals. However, after the change of variables (6.7.227), the system (6.7.239)–(6.7.247) splits as follows: ⎧  ξ = −wn−1 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 , wn−1 = sin ξ cos ξ − wn−2 (6.7.260) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ , = wn−2 wn−1 ⎩wn−2 sin ξ ⎧ ⎪ 1 + ws2 cos ηs ⎨w = d (w , s n−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) s ws sin ηs (6.7.261) ⎪ ⎩η  = d (w , . . . , w ; ξ, η , . . . , η ), s = 1, . . . , n − 3, s n−1 1 1 n−2 s  = dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ), ηn−2

(6.7.262)

where the conditions (6.7.231) hold. The system (6.7.260)–(6.7.262) is studied on the tangent bundle (6.7.233) of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. We see that the independent third-order subsystem (6.7.260) (which can be considered separately on its own three-dimensional manifold) and n − 3 independent second-order subsystems (6.7.261) (after the change of independent variable) can be substituted into the system (6.7.260)–(6.7.262) of the order 3 + 2(n − 3) + 1 = 2(n − 1), and also Eq. (6.7.262) for ηn−2 is separated (due to the fact that the variable ηn−2 is cyclic). 355

Thus, to integrate the system (6.7.260)–(6.7.262) completely, it suffices to specify two independent first integrals of system (6.7.260), n − 3 first integrals of the systems (6.7.261), and an additional first integral that “attaches” Eq. (6.7.262) (i.e., only n first integrals). Remark 6.7.7. Rewriting the first integrals (6.7.254)–(6.7.259) in the variables w1 , . . . , wn−1 by virtue of (6.7.227), we obtain 2 2 wn−2 + wn−1 + sin2 ξ = C1 = const, wn−2 sin ξ Θ2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = wn−2 sin ξ = C2 = const, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, Θs+2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = sin ηs Θn (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = Cn = const .

Θ1 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) =

(6.7.263) (6.7.264) (6.7.265) (6.7.266)

Thus, two independent first integrals (6.7.263) and (6.7.264) are sufficient to integrate the system (6.7.260), the n − 3 first integrals (4.6.68) are sufficient to integrate the independent first-order equations 1 + ws2 cos ηs dws = , s = 1, . . . , n − 3, (6.7.267) dηs ws sin ηs which is equivalent to the systems (6.7.261) after the change of independent variable, and, finally, the first integral (6.7.266) is sufficient “to attach” Eq. (6.7.262). We have proved the following theorem. Theorem 6.7.15. The system (6.7.239)–(6.7.247) of the order 2(n−1) possesses the sufficient number (namely, n) of the independent first integrals. 6.7.6. Complete list of first integrals for any finite n. Now we turn to the integration of the desired system (6.7.10)–(6.7.18) of the order 2(n − 1) (without any simplifications, i.e., in the presence of all coefficients). Similarly, to integrate the system (6.7.10)–(6.7.18) of the order 2(n − 1) completely, we need, in general, 2n − 3 independent first integrals. However, after the change of variables (6.7.227) the system (6.7.10)–(6.7.18) splits as follows: ⎧  ξ = −wn−1 − b∗ sin ξ, ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 , wn−1 = sin ξ cos ξ − wn−2 (6.7.268) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ , = wn−2 wn−1 ⎩wn−2 sin ξ ⎧ ⎪ 1 + ws2 cos ηs ⎨w = d (w , . . . , w ; ξ, η , . . . , η ) , s n−1 1 1 n−2 s ws sin ηs (6.7.269) ⎪ ⎩η  = d (w s = 1, . . . , n − 3, s n−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ), s  = dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ), ηn−2

(6.7.270)

where the conditions (6.7.231) hold. The system (6.7.268)–(6.7.270) is studied on the tangent bundle (6.7.233) of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. We see that the independent third-order subsystem (6.7.268) (which can be considered separately on its own three-dimensional manifold), n − 3 independent second-order subsystems (6.7.269) (after the change of independent variable) can be substituted into the system (6.7.268)–(6.7.270) of the order 3 + 2(n − 3) + 1 = 2(n − 1), and also Eq. (6.7.270) for ηn−2 is separated (due to the fact that the variable ηn−2 is cyclic). 356

Thus, to integrate the system (6.7.268)–(6.7.270) completely, it suffices to specify two independent first integrals of system (6.7.268), n − 3 first integral of the systems (6.7.269), and an additional first integral that “attaches” Eq. (6.7.270) (i.e., only n first integrals). First, we compare the third-order system (6.7.268) with the nonautonomous second-order system ⎧ 2 sin ξ cos ξ − wn−2 cos ξ/ sin ξ dwn−1 ⎪ ⎪ = , ⎨ dξ −wn−1 − b∗ sin ξ (6.7.271) ⎪ w dw w cos ξ/ sin ξ n−2 n−2 n−1 ⎪ ⎩ = . dξ −wn−1 − b∗ sin ξ Using the substitution τ = sin ξ, we rewrite system (6.7.271) in the algebraic form: ⎧ 2 τ − wn−2 /τ dwn−1 ⎪ ⎪ = , ⎨ dτ −wn−1 − b∗ τ ⎪ ⎪ ⎩ dwn−2 = wn−2 wn−1 /τ . dτ −wn−1 − b∗ τ

(6.7.272)

Further, if we introduce the uniform variables by the formulas wn−1 = u2 τ,

wn−2 = u1 τ,

we reduce system (6.7.272) to the following form: ⎧ 1 − u21 du2 ⎪ ⎪ + u2 = , ⎨τ dτ −u2 − b∗ ⎪ u1 u2 du ⎪ ⎩τ 1 + u1 = , dτ −u2 − b∗ which is equivalent to

⎧ 1 − u21 + u22 − bu2 du2 ⎪ ⎪ = , ⎨τ dτ −u2 − b∗ ⎪ 2u u − bu1 du ⎪ ⎩τ 1 = 1 2 . dτ −u2 − b∗

(6.7.273)

(6.7.274)

(6.7.275)

We compare the second-order system (6.7.275) with the nonautonomous first-order equation 1 − u21 + u22 + b∗ u2 du2 = , du1 2u1 u2 + b∗ u1 which can be easily reduced to the exact differential equation  2 u2 + u21 + b∗ u2 + 1 = 0. d u1

(6.7.276)

(6.7.277)

Therefore, Eq. (6.7.276) has the first integral u22 + u21 + b∗ u2 + 1 = C1 = const, u1

(6.7.278)

which in the old variables has the form Θ1 (wn−1 , wn−2 ; ξ) =

2 2 wn−1 + wn−2 + b∗ wn−1 sin ξ + sin2 ξ = C1 = const . wn−2 sin ξ

(6.7.279)

357

Remark 6.7.8. We consider the system (6.7.268) with variable dissipation with zero mean (see [161, 168]), which becomes conservative for b∗ = 0: ⎧  ξ = −wn−1 , ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 , wn−1 = sin ξ cos ξ − wn−2 (6.7.280) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ = wn−2 wn−1 . ⎩wn−2 sin ξ It has two analytical first integrals of the form 2 2 wn−1 + wn−2 + sin2 ξ = C1∗ = const,

(6.7.281)

wn−2 sin ξ = C2∗ = const .

(6.7.282)

It is obvious that the ratio of the first integrals (6.7.281) and (6.7.282) is also a first integral of system (6.7.280). However, for b∗ = 0 both functions 2 2 + wn−2 + b∗ wn−1 sin ξ + sin2 ξ wn−1

(6.7.283)

and (6.7.282) are not first integrals of the system (6.7.268), but their ratio (i.e., the ratio of the functions (6.7.283) and (6.7.282)) is a first integral of system (6.7.268) for any b∗ . In the sequel, we find an obvious form of the additional first integral of the third-order sys 0 as follows: tem (6.7.268). For this, we first transform the invariant relation (6.7.278) for u1 = 2  2  b∗ C1 b2 + C12 + u1 − = ∗ − 1. (6.7.284) u2 + 2 2 4 We see that the parameters of the given invariant relation must satisfy the condition b2∗ + C12 − 4 ≥ 0,

(6.7.285)

and the phase space of system (6.7.268) is stratified into a family of surfaces defined by Eq. (6.7.284) in the coordinates u1 and u2 . Thus, by virtue of relation (6.7.278), the first equation of system (6.7.275) has the form τ

2(1 + b∗ u2 + u22 ) − C1 U1 (C1 , u2 ) du2 = , dτ −u2 − b∗

(6.7.286)

where

$ 1 {C1 ± C12 − 4(u22 + b∗ u2 + 1)}, (6.7.287) 2 and the integration constant C1 is chosen from the condition (6.7.285). Therefore, the quadrature for the search for an additional first integral of the system (6.7.268) has the form & & (−b∗ − u2 )du2 dτ ( = . (6.7.288) 2 τ 2(1 + b∗ u2 + u2 ) − C1 {C1 ± C12 − 4(u22 + b∗ u2 + 1)}/2 U1 (C1 , u2 ) =

Obviously, the left-hand side up to an additive constant is equal to ln | sin ξ|.

(6.7.289)

If u2 + 358

b∗ = r1 , 2

b21 = b2∗ + C12 − 4,

(6.7.290)

then the right-hand side of Eq. (6.7.288) has the form & & d(b21 − 4r12 ) 1 dr1 ( ( −b − 2 2 2 2 2 2 4 (b1 − 4r1 ) ± C1 b1 − 4r1 (b1 − 4r1 ) ± C1 b21 − 4r12 (   b 1  b21 − 4r12  = − ln  ± 1 ± I1 , (6.7.291)  2 2  C1 where

& (

I1 =

$

dr3 b21 − r32 (r3 ± C1 )

,

r3 =

b21 − 4r12 .

(6.7.292)

In the calculation of the integral (6.7.292), the following three cases are possible. I. b∗ > 2. (   b2 − 4 + ( b2 − r 2 1 C1   ∗ 1 3 ±( I1 = − ( ln   r3 ± C1 2 b2∗ − 4  b2∗ − 4  (   b2 − 4 − (b2 − r 2  1 C   1 ∗ 1 3 ln  ∓( + (  + const . (6.7.293) 2 2  r3 ± C1 2 b∗ − 4 b∗ − 4  II. b∗ < 2.

1 ±C1 r3 + b21 + const . I1 = ( arcsin b1 (r3 ± C1 ) 4 − b2∗

III. b∗ = 2.

(6.7.294)

(

I1 = ∓

b21 − r32 + const . C1 (r3 ± C1 )

(6.7.295)

When we return to the variable r1 =

wn−1 b∗ + , sin ξ 2

(6.7.296)

we obtain the final form for the value I1 : I. b∗ > 2.  (   b2 − 4 ± 2r  1 C   1 1 ( ln  ( 2∗ ± I1 = − (  2 2 2 2 b∗ − 4  b1 − 4r1 ± C1 b∗ − 4   (   b2 − 4 ∓ 2r  C 1   1 ( 1  + const . (6.7.297) ln  ( 2∗ ∓ + ( 2 b2∗ − 4  b1 − 4r12 ± C1 b2∗ − 4  II. b∗ < 2.

( ±C1 b21 − 4r12 + b21 ( arcsin + const . I1 = ( 4 − b2∗ b1 ( b21 − 4r12 ± C1 ) 1

III. b∗ = 2. I1 = ∓

2r1 ( + const . C1 ( b21 − 4r12 ± C1 )

(6.7.298)

(6.7.299)

Thus, we have found an additional first integral for the third-order system (6.7.268), i.e., we have a complete set of first integrals that are transcendental functions of the phase variables. Remark 6.7.9. In the expression of the found first integral, we must formally substitute the left-hand side of the first integral (6.7.278) instead of C1 . 359

Then the obtained additional first integral has the following structure:  wn−1 wn−2 Θ2 (wn−1 , wn−2 ; ξ) = G2 sin ξ, , = C2 = const . sin ξ sin ξ

(6.7.300)

Thus, we have found two first integrals (6.7.279) and (6.7.300) of the independent third-order system (6.7.268). To integrate it completely, it suffices to find n−3 first integrals for the systems (6.7.269) and an additional first integral that “attaches” Eq. (6.7.270). Indeed, the desired first integrals coincide with the first integrals (6.7.265) and (6.7.266), namely, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, (6.7.301) Θs+2 (ws ; ηs ) = sin ηs Cn−1 cos ηn−3 = Cn = const . (6.7.302) Θn (wn−3 , wn−4 ; ηn−4 , ηn−3 , ηn−2 ) = ηn−2 ± arctg $ 2 2 2 Cn−2 sin ηn−3 − Cn−1 In this case, on the left-hand side of Eq. (6.7.302), we must substitute the first integrals (6.7.301) for s = n − 4 and s = n − 3 instead of Cn−2 and Cn−1 . Theorem 6.7.16. The system (6.7.268)–(6.7.270) of the order 2(n − 1) possesses a sufficient number (namely, n) of independent first integrals (6.7.279), (6.7.300), (6.7.301), and (6.7.302). Therefore, in the considered case, the system of dynamical equations (6.7.10)–(6.7.18) has n first integrals expressed by the relations (6.7.279), (6.7.300), (6.7.301), and (6.7.302), which are transcendental functions of the phase variables (in the sense of the complex analysis) and are expressed as a finite combination of elementary functions (in this case, we use the expressions (6.7.288)–(6.7.299)). Theorem 6.7.17. The three sets of relations (6.3.1), (6.4.3), and (6.5.9) under the conditions (6.3.6)–(6.3.8), (6.7.1), and (6.7.5) possess n first integrals (a complete set), which are transcendental functions (in the sense of complex analysis) and are expressed as finite combinations of elementary functions. 6.7.7. Topological analogies. Now we present two groups of analogies related to the system (6.6.1), which describes the motion of a free body in the presence of a tracking force. The first group of analogies deals with the case of the presence the nonintegrable constraint (6.6.12) in the system. In this case the dynamical part of the motion equations under certain conditions is reduced to the system (6.6.31). Under conditions (6.7.1) and (6.7.5) the system (6.6.31) has the form α = −Zn−1 + b sin α, cos α , sin α cos α cos α cos β1  2 + (Z12 + . . . + Zn−3 = Zn−2 Zn−1 ) , Zn−2 sin α sin α sin β1 cos α cos α cos β1  − Zn−3 Zn−2 = Zn−3 Zn−1 Zn−3 sin α sin α sin β1 cos α 1 cos β2 2 − (Z12 + . . . + Zn−4 ) , sin α sin β1 sin β2 .......................................... n−2   cos α cos β s−1 (−1)s+1 Zn−s , Z1 = Z1 sin α sin β1 . . . sin βs−1 s=1 cos α , β1 = Zn−2 sin α  2 = sin α cos α − (Z12 + . . . + Zn−2 ) Zn−1

360

(6.7.303) (6.7.304) (6.7.305)

(6.7.306)

(6.7.307) (6.7.308)

cos α , (6.7.309) sin α sin β1 .......................................... cos α  βn−3 = (−1)n Z2 , (6.7.310) sin α sin β1 . . . sin βn−4 cos α  = (−1)n+1 Z1 , (6.7.311) βn−2 sin α sin β1 . . . sin βn−3 if we introduce the dimensionless parameter, the variables, and the differentiation analogously to (6.7.6):  AB b = σn0 , n20 = , zk = n0 vZk , k = 1, . . . , n − 1, · = n0 v  . (6.7.312) (n − 2)I2 β2 = −Zn−3

In particular, for n = 5 we obtain the following eighth-order system: α = −Z4 + b sin α, Z4 Z3 Z2 Z1 β1 β2 β3

(6.7.313)

cos α = sin α cos α − (Z12 + Z22 + Z32 ) , sin α cos α cos β1 cos α = Z3 Z4 + (Z12 + Z22 ) , sin α sin α sin β1 cos α cos α cos β1 cos α 1 cos β2 = Z2 Z4 − Z2 Z3 − Z12 , sin α sin α sin β1 sin α sin β1 sin β2 cos α cos α cos β1 cos α 1 cos β2 = Z1 Z4 − Z1 Z3 + Z1 Z2 , sin α sin α sin β1 sin α sin β1 sin β2 cos α , = Z3 sin α cos α = −Z2 , sin α sin β1 cos α = Z1 . sin α sin β1 sin β2

(6.7.314) (6.7.315) (6.7.316) (6.7.317) (6.7.318) (6.7.319) (6.7.320)

Theorem 6.7.18. System (6.7.303)–(6.7.311) (for the case of a free body) is equivalent to the system (6.7.10)–(6.7.18) (for the case of a fixed pendulum). Indeed, it is sufficient to substitute ξ = α,

η1 = β1 ,

...,

ηn−2 = βn−2 ,

b∗ = −b,

(6.7.321)

and also to compare the variables Zk ↔ −Zk , k = 1, . . . , n − 1. To integrate completely the system (6.7.303)–(6.7.311) of the order 2(n − 1), in general, we need 2n − 3 independent first integrals (in particular, to integrate completely the eighth-order system (6.7.313)–(6.7.320), in general, we need seven independent first integrals). However, after the change of variables ⎞ ⎛ ⎞ ⎛ wn−1 Zn−1 ⎜ wn−2 ⎟ ⎜ Zn−2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ... ⎟ → ⎜ ... ⎟, ⎟ ⎜ ⎟ ⎜ ⎝ w2 ⎠ ⎝ Z2 ⎠ Z1 w1 $ Z2 Z3 2 , wn−3 = , wn−4 = ( 2 , . . . , (6.7.322) wn−1 = Zn−1 , wn−2 = Z12 + . . . + Zn−2 Z1 Z1 + Z22 w2 = $

Zn−3 2 Z12 + . . . + Zn−4

,

w1 = $

Zn−2 2 Z12 + . . . + Zn−3

,

361

the system (6.7.303)–(6.7.311) splits as follows: ⎧  α = −wn−1 + b sin α, ⎪ ⎪ ⎪ ⎨  cos α 2 wn−1 = sin α cos α − wn−2 , sin α ⎪ ⎪ cos α ⎪ ⎩w  , n−2 = wn−2 wn−1 sin α ⎧ ⎪ ⎨w = d (w , . . . , w ; α, β , . . . , β

1 + ws2 cos βs , ws sin βs ⎪ ⎩β  = d (w s = 1, . . . , n − 3, s n−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), s s

s

n−1

1

1

n−2 )

 = dn−2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), βn−2

(6.7.323)

(6.7.324) (6.7.325)

where cos α , sin α cos α d2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = −Zn−3 (wn−1 , . . . , w1 ) , sin α sin β1 ............................................................ d1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = Zn−2 (wn−1 , . . . , w1 )

(6.7.326)

dn−2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = (−1)n+1 Z1 (wn−1 , . . . , w1 ) cos α , × sin α sin β1 . . . sin βn−3 in this case Zk = Zk (wn−1 , . . . , w1 ),

k = 1, . . . , n − 2,

(6.7.327)

are the functions, by virtue of change (6.7.322). In particular, for n = 5 the system (6.7.313)–(6.7.320) splits as follows: ⎧  α = −w4 + b sin α, ⎪ ⎪ ⎪ ⎨  cos α , w4 = sin α cos α − w32 sin α ⎪ ⎪ ⎪ ⎩w = w3 w4 cos α , 3 sin α ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w2 cos β2 , 2 4 3 2 1 1 2 3 2 w2 sin β2 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w1 cos β1 , 1 4 3 2 1 1 2 3 1 w1 sin β1 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 1 4 3 2 1 1 2 3 1 β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ), 362

(6.7.328)

(6.7.329)

(6.7.330) (6.7.331)

where cos α w1 w3 cos α = ±( , sin α 1 + w12 sin α cos α d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = −Z2 (w4 , w3 , w2 , w1 ) sin α sin β1 w2 w3 cos α ( = ∓( , 2 2 sin α sin β1 1 + w1 1 + w2 cos α d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = Z1 (w4 , w3 , w2 , w1 ) sin α sin β1 sin β2 w3 cos α ( = ±( , 2 2 1 + w1 1 + w2 sin α sin β1 sin β2 d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = Z3 (w4 , w3 , w2 , w1 )

(6.7.332)

in this case Zk = Zk (w4 , w3 , w2 , w1 ),

k = 1, 2, 3,

(6.7.333)

are the functions, by virtue of change (6.7.322). The system (6.7.323)–(6.7.325) is studied on the tangent bundle T∗ Sn−1 {(wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) ∈ R2(n−1) : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π} (6.7.334) of the (n − 1)-dimensional sphere Sn−1 {(α, β1 , . . . , βn−2 ) ∈ Rn−1 : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π}. In particular, the system (6.7.328)–(6.7.331) is studied on the tangent bundle T∗ S4 {(w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) ∈ R8 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}

(6.7.335)

of the four-dimensional sphere S4 {(α, β1 , β2 , β3 ) ∈ R4 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}. We see that the independent third-order subsystem (6.7.323) (which can be considered separately on its own three-dimensional manifold) and n − 3 independent second-order subsystem (6.7.324) (after the change of independent variable) can be substituted into the sixth-order system (6.7.323)–(6.7.325) of the order 2(n − 1), and also Eq. (6.7.325) on βn−2 is separated (due to the fact that the variable ηn−2 is cyclic). In particular, we see that the independent third-order subsystem (6.7.328) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.7.329) and (6.7.330) (after the change of independent variable) can be substituted into the fourth-order system (6.7.328)–(6.7.331), and also Eq. (6.7.331) on β3 is separated (due to the fact that the variable β3 is cyclic). Thus, to integrate completely the system (6.7.323)–(6.7.325), it suffices to specify two independent first integrals of system (6.7.323), one by one first integral of systems (6.7.324) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.7.325) (i.e., only n). In particular, to integrate completely the system (6.7.328)–(6.7.331), it suffices to specify two independent first integrals of system (6.7.328), one by one first integral of systems (6.7.329) and (6.7.330), and an additional first integral that “attaches” Eq. (6.7.331) (i.e., only five). Corollary 6.7.1. 1. The attack angle α and the angles β1 , . . . , βn−2 in the case of a free body are equivalent to the angles of a body’s deviation ξ and η1 , . . . , ηn−2 , respectively, of a fixed pendulum. 2. The distance σ = CD for a free body corresponds to the length of a holder l = OD of a fixed pendulum. 363

3. The first integrals of the system (6.7.323)–(6.7.325) can be automatically obtained through Eqs. (6.7.279), (6.7.300), (6.7.301), and (6.7.302) after the substitutions (6.7.321) (see also [199]): 2 2 wn−1 + wn−2 − bwn−1 sin α + sin2 α = C1 = const, wn−2 sin α   wn−1 wn−2 , = C2 = const, Θ2 (wn−1 , wn−2 ; α) = G sin α, sin α sin α ( 1 + ws2 = Cs+2 = const, s = 1, . . . , n − 3, Θs+2 (ws ; βs ) = sin βs Cn−1 cos βn−3 Θn (wn−3 , wn−4 ; βn−4 , βn−3 , βn−2 ) = βn−2 ± arctg $ 2 2 Cn−2 sin2 βn−3 − Cn−1

Θ1 (wn−1 , wn−2 ; α) =

= Cn = const;

(6.7.336) (6.7.337) (6.7.338)

(6.7.339)

in this case, on the left-hand side of Eq. (6.7.339), we must substitute instead of Cn−2 , Cn−1 the first integrals (6.7.338) for s = n − 4, n − 3. The second group of analogies deals with the case of motion with constant velocity of the center of mass of a body, i.e., when the property (6.6.39) holds. In this case the dynamical part of the motion equations under certain conditions is reduced to the system (6.6.45)–(6.6.51). Then, under the conditions (6.6.39), (6.7.1), (6.7.5), and (6.7.312), the reduced dynamical part of the motion equations (system (6.6.46)–(6.6.54)) has the form of the analytical system n−1    α = −Zn−1 + b Zs2 sin α + b sin α cos2 α, (6.7.340) s=1

 Zn−1

 Zn−2

n−2 



n−1   cos α = sin α cos α − Zs2 + bZn−1 Zs2 cos α − bZn−1 sin2 α cos α, sin α s=1 s=1 n−3  n−1    cos α cos α cos β1 2 = Zn−2 Zn−1 + Zs + bZn−2 Zs2 cos α sin α sin α sin β 1 s=1 s=1 − bZn−2 sin2 α cos α,

 = Zn−3 Zn−1 Zn−3

n−4 



cos α cos α cos β1 cos α 1 cos β2 − Zn−3 Zn−2 − Zs2 sin α sin α sin β1 sin α sin β1 sin β2 s=1 n−1   + bZn−3 Zs2 cos α − bZn−3 sin2 α cos α,

(6.7.341)

(6.7.342)

(6.7.343)

s=1

.......................................... n−2   cos α cos β s−1 (−1)s+1 Zn−s Z1 = Z1 sin α s=1 sin β1 . . . sin βs−1 n−1   Zs2 cos α − bZ1 sin2 α cos α, + bZ1

(6.7.344)

s=1

cos α , = Zn−2 sin α cos α , β2 = −Zn−3 sin α sin β1 .......................................... β1

364

(6.7.345) (6.7.346)

cos α , sin α sin β1 . . . sin βn−4 cos α  βn−2 = (−1)n+1 Z1 . sin α sin β1 . . . sin βn−3 In this case, we choose the constant n1 as follows:  βn−3 = (−1)n Z2

n1 = n0 . In particular, for n = 5 we obtain the following eighth-order system:   α = −Z4 + b Z12 + Z22 + Z32 + Z42 sin α + b sin α cos2 α,

(6.7.347) (6.7.348)

(6.7.349) (6.7.350)

  cos α Z4 = sin α cos α − Z12 + Z22 + Z32 sin α   + bZ4 Z12 + Z22 + Z32 + Z42 cos α − bZ4 sin2 α cos α, (6.7.351) Z3 = Z3 Z4

 cos α cos β1 cos α  2 + Z1 + Z22 sin α sin α sin β1   + bZ3 Z12 + Z22 + Z32 + Z42 cos α − bZ3 sin2 α cos α, (6.7.352)

Z2 = Z2 Z4

cos α cos α cos β1 cos α 1 cos β2 − Z2 Z3 − Z12 sin α sin α sin β1 sin α sin β1 sin β2  2  + bZ2 Z1 + Z22 + Z32 + Z42 cos α − bZ2 sin2 α cos α, (6.7.353)

Z1 = Z1 Z4

cos α cos α cos β1 cos α 1 cos β2 − Z1 Z3 + Z1 Z2 sin α sin α sin β1 sin α sin β1 sin β2  2  + bZ1 Z1 + Z22 + Z32 + Z42 cos α − bZ1 sin2 α cos α, (6.7.354) cos α , sin α cos α , β2 = −Z2 sin α sin β1 cos α . β3 = Z1 sin α sin β1 sin β2

β1 = Z3

(6.7.355) (6.7.356) (6.7.357)

To integrate completely the system (6.7.340)–(6.7.348) of the order 2(n − 1), in general, we need 2n−3 independent first integrals. However, after the change of variables (6.7.322) the system (6.7.340)– (6.7.348) splits as follows: ⎧  2 2 + wn−1 ) sin α + b sin α cos2 α, α = −wn−1 + b(wn−2 ⎪ ⎪ ⎪ ⎨  cos α 2 2 2 + bwn−1 (wn−2 + wn−1 ) cos α − bwn−1 sin2 α cos α, wn−1 = sin α cos α − wn−2 (6.7.358) sin α ⎪ ⎪ cos α ⎪ 2 2 ⎩w  + bwn−2 (wn−2 + wn−1 ) cos α − bwn−2 sin2 α cos α, n−2 = wn−2 wn−1 sin α ⎧ ⎪ 1 + ws2 cos βs ⎨w = d (w , s n−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) s ws sin βs (6.7.359) ⎪ ⎩β  = d (w , . . . , w ; α, β , . . . , β ), s = 1, . . . , n − 3, s n−1 1 1 n−2 s  = dn−2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), βn−2

(6.7.360)

where the conditions (6.7.326) hold. 365

In particular, for n = 5 the system (6.7.350)–(6.7.357) splits as follows: ⎧  α = −w4 + b(w32 + w42 ) sin α + b sin α cos2 α, ⎪ ⎪ ⎪ ⎨  cos α + bw4 (w32 + w42 ) cos α − bw4 sin2 α cos α, w4 = sin α cos α − w32 sin α ⎪ ⎪ ⎪ ⎩w = w w cos α + bw (w2 + w2 ) cos α − bw sin2 α cos α, 3 4 3 3 3 3 4 sin α ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w2 cos β2 , 2 4 3 2 1 1 2 3 2 w2 sin β2 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w1 cos β1 , 1 4 3 2 1 1 2 3 1 w1 sin β1 ⎪  ⎩β = d (w , w , w , w ; α, β , β , β ), 1 4 3 2 1 1 2 3 1 β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(6.7.361)

(6.7.362)

(6.7.363) (6.7.364)

where the conditions (6.7.332) hold. The system (6.7.358)–(6.7.360) is studied on the tangent bundle (6.7.334) of the (n − 1)-dimensional sphere Sn−1 {(α, β1 , . . . , βn−2 ) ∈ Rn−1 : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π}. In particular, the system (6.7.361)–(6.7.364) is studied on the tangent bundle (6.7.335) of the fourdimensional sphere S4 {(α, β1 , β2 , β3 ) ∈ R4 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}. We see that the independent third-order subsystem (6.7.358) (which can be considered separately on its own three-dimensional manifold) and n−3 independent second-order subsystems (6.7.359) (after the change of independent variable) can be substituted into the system (6.7.358)–(6.7.360) of the order 2(n − 1), and also Eq. (6.7.360) on βn−2 is separated (due to the fact that the variable βn−2 is cyclic). In particular, we see that the independent third-order subsystem (6.7.361) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.7.362) and (6.7.363) (after the change of independent variable) can be substituted into the eighth-order system (6.7.361)–(6.7.364), and also Eq. (6.7.364) on β3 is separated (due to the fact that the variable β3 is cyclic). Thus, to integrate completely the system (6.7.358)–(6.7.360), it suffices to specify two independent first integrals of system (6.7.358), one by one the first integral of systems (6.7.359) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.7.360) (i.e., only n). In particular, to integrate completely the system (6.7.361)–(6.7.364), it suffices to specify two independent first integrals of system (6.7.361), one by one the first integral of the systems (6.7.362) and (6.7.363), and an additional first integral that “attaches” Eq. (6.7.364) (i.e., only five). If the problem on the first integrals of the system (6.7.303)–(6.7.311) (or (6.7.323)–(6.7.325)) is solved using Corollary 6.7.1, the same problem for the system (6.7.340)–(6.7.348) (or (6.7.358)– (6.7.360)) can be solved by Theorem 6.7.19 below. First, we note that one of the first integrals of the system (6.7.358) has the following form (see [199]): Θ1 (wn−1 , wn−2 ; α)

2 2 wn−1 + wn−2 − bwn−1 sin α + sin2 α = = C1 = const . wn−2 sin α

(6.7.365)

In the sequel, we study an additional first integral of the third-order system (6.7.358) using, in this case, the first integral (6.7.365). For this we introduce the following notation and new variables: τ = sin α, 366

wn−1 = u2 τ,

wn−2 = u1 τ,

p=

1 . τ2

(6.7.366)

Then the problem on the explicit form of the desired first integral reduces to solving the linear inhomogeneous equation 2(u2 − b)p + 2b(1 − U12 (C1 , u2 ) − u22 ) dp = , du2 1 − bu2 + u22 − U12 (C1 , u2 )   $ 1 2 2 U1 (C1 , u2 ) = C1 ± C1 − 4(u2 − bu2 + 1) ; 2 in this case, an additive constant C1 can be chosen as follows: b2 + C12 − 4 ≥ 0.

(6.7.367)

(6.7.368)

The last fact means that we can find another transcendental first integral in the explicit form (i.e., as a finite combination of quadratures). Here, the general solution of Eq. (6.7.367) depends on an arbitrary constant C2 . We omit the calculation but note that the general solution of the linear homogeneous equation obtained from (6.7.367) even in the particular case b = C1 = 2 has the following solution: -% . ( 2 ( 1 ∓ 1 − (u − 1) 2 ( p = p0 (u2 ) = C[ 1 − (u2 − 1)2 ± 1] exp , C = const . (6.7.369) 1 ± 1 − (u2 − 1)2 Then the desired additional first integral has the following structural form:  wn−1 wn−2  , = C2 = const; (6.7.370) Θ2 (wn−1 , wn−2 ; α) = G sin α, sin α sin α in this case, we use the notation and substitutions (6.7.366). Thus, we have found two first integrals (6.7.365), (6.7.370) of the independent third-order system (6.7.358). To integrate it completely, it suffices to find one by one the first integral for the systems (6.7.359) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.7.360). Indeed, the desired first integrals coincide with the first integrals (6.7.338) and (6.7.339), namely, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, (6.7.371) Θs+2 (ws ; βs ) = sin βs Cn−1 cos βn−3 = Cn = const, (6.7.372) Θn (wn−3 , wn−4 ; βn−4 , βn−3 , βn−2 ) = βn−2 ± arctg $ 2 2 Cn−2 sin2 βn−3 − Cn−1 in this case, on the left-hand side of Eq. (6.7.372), we must substitute instead of Cn−2 , Cn−1 the first integrals (6.7.371) for s = n − 4, n − 3. Theorem 6.7.19. The n first integrals (6.7.365), (6.7.370), (6.7.371), and (6.7.372) of the system (6.7.358)–(6.7.360) are transcendental functions of the phase variables and are expressed as a finite combination of elementary functions. Theorem 6.7.20. The n first integrals (6.7.365), (6.7.370), (6.7.371), and (6.7.372) of the system (6.7.358)–(6.7.360) are equivalent to n first integrals (6.7.336), (6.7.337), (6.7.338), and (6.7.339) of the system (6.7.323)–(6.7.325). Indeed, the tuples of the first integrals (6.7.365), (6.7.336), (6.7.371), (6.7.338) and (6.7.372), (6.7.339) coincide if we substitute b = −b∗ . Finally, we need to identify the phase variables wk , k = 1, . . . , n − 1, for the system (6.7.358)–(6.7.360) with the phase variables wk , k = 1, . . . , n − 1, for the system (6.7.323)–(6.7.325). Because of their cumbersome character, similar arguments concerning the couples of the first integrals (6.7.370) and (6.7.337) are not represented. Thus, we have the following topological and mechanical analogies in the sense explained above. (1) Motion of a fixed multi-dimensional physical pendulum on a (generalized) spherical hinge in a flowing medium (nonconservative force fields). 367

(2) Free motion of a multi-dimensional rigid body in a nonconservative force field under a tracing force (in the presence of a nonintegrable constraint). (3) Composite motion of a multi-dimensional rigid body rotating about its center of mass, which moves rectilinearly and uniformly, in a nonconservative force field. On more general topological analogues, see also [2, 11, 21, 24, 37, 54]. 6.8.

Case where the Moment of Nonconservative Forces Depends on the Angular Velocity

6.8.1. Dependence on the angular velocity. This section is devoted to dynamics of a multidimensional rigid body in the multi-dimensional space En . Since this subsection is devoted to the study of the case of motion where the moment of forces depends on the tensor of angular velocity, we introduce this dependence in the general case. Let x = (x1N , . . . , xnN ) be the coordinates of the point N of application of a nonconservative force (interaction with a medium) on the (n − 1)-dimensional disk D n−1 , and Q = (Q1 , . . . , Qn ) be the components independent of the angular velocity. We introduce only the linear dependence of the functions (x1N , . . . , xnN ) on the tensor of angular velocity Ω since the introduction of this dependence itself is not a priori obvious (see [41, 42, 199]). Thus, we accept the following dependence: x = Q + R,

(6.8.1)

where R = (R1 , . . . , Rn ) is a vector-valued function containing the tensor of angular velocity Ω. Here, the dependence of the function R on the tensor of angular velocity is gyroscopic: ⎛ ⎛ ⎞ ⎞ h1 R1 ⎟ ⎜ R2 ⎟ 1 ⎜ ⎜ h2 ⎟ ⎜ ⎟ (6.8.2) R = ⎜ .. ⎟ = − Ω ⎜ .. ⎟ , vD ⎝ . ⎠ ⎝ . ⎠ Rn hn where (h1 , . . . , hn ) are certain positive parameters. Now, for our problem, since x1N = xN ≡ 0, we have x2N = Q2 − h1

ωrn−1 , vD

x3N = Q3 + h1

ωrn−2 , vD

...,

xnN = Qn + (−1)n+1 h1

ω r1 . v

(6.8.3)

Thus, the function rN is selected in the following form (the disk D n−1 is defined by the equation x1N ≡ 0): ⎛ ⎞ 0 ⎜ x2N ⎟ 1 ⎜ ⎟ Ωh, (6.8.4) rN = ⎜ . ⎟ = R(α)iN − vD ⎝ .. ⎠ xnN where

⎛ iN = iv

π 2

 , β1 , . . . , βn−2 ,

⎜ ⎜ h=⎜ ⎝

h1 h2 .. . hn

(see (6.1.6) and (6.4.2)). 368

⎞ ⎟ ⎟ ⎟, ⎠

Ω ∈ so(n)

(6.8.5)

⎛

In our case

iN

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

0 cos β1 sin β1 cos β2 ... sin β1 . . . sin βn−3 cos βn−2 sin β1 . . . sin βn−2

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(6.8.6)

Thus, the relations x2N = R(α) cos β1 − h1

ωrn−1 , vD

x3N = R(α) sin β1 cos β2 + h1

xn−1,N = R(α) sin β1 . . . sin βn−3 cos βn−2 + (−1)n h1

ω r2 , v

ωrn−2 , vD

..., (6.8.7)

ω r1 , v hold, which show that an additional dependence of the damping (or accelerating in some domains of the phase space) moment of the nonconservative forces is also present in the system considered (i.e., the moment depends on the tensor of angular velocity). Thus, for the construction of the force field, we use the pair of dynamical functions R(α), s(α); the information about them is of a qualitative nature. Similarly to the choice of the Chaplygin analytical functions (see [122, 123]), we take the dynamical functions s and R as follows: xnN = R(α) sin β1 . . . sin βn−2 + (−1)n+1 h1

R(α) = A sin α, 6.8.2.

s(α) = B cos α,

A, B > 0.

(6.8.8)

Reduced systems.

Theorem 6.8.1. The simultaneous equations (6.3.1), (6.4.3), and (6.5.9) under the conditions (6.3.6)–(6.3.8), (6.8.4), and (6.8.8) can be reduced to the dynamical system on the tangent bundle (6.1.5) of the (n − 1)-dimensional sphere (6.1.4). Indeed, if we introduce the dimensionless parameters and differentiation by the formulas  AB h1 B , H1∗ = , · = n0 v∞  , b∗ = ln0 , n20 = (n − 2)I2 (n − 2)I2 n0 then the obtained equations have the following form: ξ  + (b∗ − H1∗ )ξ  cos ξ + sin ξ cos ξ− −

η12

+

η22

2

sin η1 +

η32

2

2

sin η1 sin η2 + . . . +

2 ηn−2 sin2 η1 . . . sin2 ηn−3

η1 + (b∗ − H1∗ )η1 cos ξ + ξ  η1

1 + cos2 ξ cos ξ sin ξ

η2 + (b∗ − H1∗ )η2 cos ξ + ξ  η2

1 + cos2 ξ cos η1 + 2η1 η2 cos ξ sin ξ sin η1

η3 + (b∗ − H1∗ )η3 cos ξ + ξ  η3

1 + cos2 ξ cos η1 cos η2 + 2η1 η3 + 2η2 η3 cos ξ sin ξ sin η1 sin η2

! sin ξ cos ξ

(6.8.9)

= 0,

! 2 sin2 η2 . . . sin2 ηn−3 sin η1 cos η1 = 0, − η22 + η32 sin2 η2 + η42 sin2 η2 sin2 η3 + . . . + ηn−2

! 2 − η32 + η42 sin2 η3 + η52 sin2 η3 sin2 η4 + . . . + ηn−2 sin2 η3 . . . sin2 ηn−3 sin η2 cos η2 = 0,

! 2 − η42 + η52 sin2 η4 + η62 sin2 η4 sin2 η5 + . . . + ηn−2 sin2 η4 . . . sin2 ηn−3 sin η3 cos η3 = 0, (6.8.10) 369

...................................................... 1 + cos2 ξ cos η1  + 2η1 ηn−4 cos ξ sin ξ sin η1 ! cos η n−5   2 2 + . . . + 2ηn−5 ηn−4 − ηn−3 + ηn−2 sin2 ηn−3 sin ηn−4 cos ηn−4 = 0, sin ηn−5

   ηn−4 + (b∗ − H1∗ )ηn−4 cos ξ + ξ  ηn−4

1 + cos2 ξ cos η1  + 2η1 ηn−3 cos ξ sin ξ sin η1 cos ηn−4   2 + . . . + 2ηn−4 ηn−3 − ηn−2 sin ηn−3 cos ηn−3 = 0, sin ηn−4

   ηn−3 + (b∗ − H1∗ )ηn−3 cos ξ + ξ  ηn−3

   ηn−2 + (b∗ − H1∗ )ηn−2 cos ξ + ξ  ηn−2

1 + cos2 ξ cos η1  + 2η1 ηn−2 cos ξ sin ξ sin η1 cos ηn−3   + . . . + 2ηn−3 ηn−2 = 0, sin ηn−3

b∗ > 0,

H1∗ > 0.

In particular, for n = 5 we have " # sin ξ = 0, ξ  + (b∗ − H1∗ )ξ  cos ξ + sin ξ cos ξ − η12 + η22 sin2 η1 + η32 sin2 η1 sin2 η2 cos ξ # 1 + cos2 ξ " 2 − η2 + η32 sin2 η2 sin η1 cos η1 = 0, η1 + (b∗ − H1∗ )η1 cos ξ + ξ  η1 cos ξ sin ξ 1 + cos2 ξ cos η1 + 2η1 η2 − η32 sin η2 cos η2 = 0, η2 + (b∗ − H1∗ )η2 cos ξ + ξ  η2 cos ξ sin ξ sin η1 1 + cos2 ξ cos η1 cos η2 + 2η1 η3 + 2η2 η3 = 0, b∗ > 0, H1∗ > 0. η3 + (b∗ − H1∗ )η3 cos ξ + ξ  η3 cos ξ sin ξ sin η1 sin η2 (6.8.11) After the transition from the variables z (about the variables z, see (6.5.7)) to the intermediate dimensionless variables w zk = n0 v∞ (1 + b∗ H1∗ )Zk ,

k = 1, . . . , n − 2,

zn−1 = n0 v∞ (1 + b∗ H1∗ )Zn−1 − n0 v∞ b∗ sin ξ, (6.8.12)

system (6.8.10) is equivalent to the system ξ  = (1 + b∗ H1∗ ) Zn−1 − b∗ sin ξ,  2 Zn−1 = − sin ξ cos ξ + (1 + b∗ H1∗ ) (Z12 + . . . + Zn−2 )

 = − (1 + b∗ H1∗ ) Zn−2 Zn−1 Zn−2

cos ξ + H1∗ Zn−1 cos ξ, sin ξ

(6.8.13) (6.8.14)

cos ξ sin ξ

2 ) − (1 + b∗ H1∗ ) (Z12 + . . . + Zn−3

cos ξ cos η1 + H1∗ Zn−2 cos ξ, (6.8.15) sin ξ sin η1

cos ξ cos ξ cos η1 + (1 + b∗ H1∗ ) Zn−3 Zn−2 sin ξ sin ξ sin η1 cos ξ 1 cos η2 2 + (1 + b∗ H1∗ ) (Z12 + . . . + Zn−4 ) + H1∗ Zn−3 cos ξ, (6.8.16) sin ξ sin η1 sin η2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. n−2  cos ξ cos ηs−1 (−1)s+1 Zn−s (6.8.17) + H1∗ Z1 cos ξ, Z1 = − (1 + b∗ H1∗ ) Z1 sin ξ sin η1 . . . sin ηs−1

 = − (1 + b∗ H1∗ ) Zn−3 Zn−1 Zn−3

s=1

370

cos ξ , sin ξ cos ξ , η2 = (1 + b∗ H1∗ ) Zn−3 sin ξ sin η1 .......................................... cos ξ  = (−1)n+1 (1 + b∗ H1∗ ) Z2 , ηn−3 sin ξ sin η1 . . . sin ηn−4 cos ξ  ηn−2 = (−1)n (1 + b∗ H1∗ ) Z1 , sin ξ sin η1 . . . sin ηn−3 η1 = − (1 + b∗ H1∗ ) Zn−2

(6.8.18) (6.8.19) (6.8.20) (6.8.21) (6.8.22)

on the tangent bundle T∗ Sn−1 {(Zn−1 , . . . , Z1 ; ξ, η1 , . . . , ηn−2 ) ∈ R2(n−1) : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π} (6.8.23) of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. We see that the independent subsystem (6.8.13)–(6.8.22) of the order 2(n − 1) (due to the fact that the variable ηn−2 is cyclic) can be substituted into the system (6.8.13)–(6.8.21) of the order 2(n−1)−1 and can be considered separately on its own (2n − 3)-dimensional manifold. In particular, for n = 5 we obtain the following eighth-order system: ξ  = (1 + b∗ H1∗ ) Z4 − b∗ sin ξ, cos ξ + H1∗ Z4 cos ξ, sin ξ cos ξ cos η1 cos ξ − (1 + b∗ H1∗ ) (Z12 + Z22 ) Z3 = − (1 + b∗ H1∗ ) Z3 Z4 + H1∗ Z3 cos ξ, sin ξ sin ξ sin η1 Z4 = − sin ξ cos ξ + (1 + b∗ H1∗ ) (Z12 + Z22 + Z32 )

(6.8.24) (6.8.25) (6.8.26)

Z2 = − (1 + b∗ H1∗ ) Z2 Z4

cos ξ cos ξ cos η1 + (1 + b∗ H1∗ ) Z2 Z3 sin ξ sin ξ sin η1 cos ξ 1 cos η2 + (1 + b∗ H1∗ ) Z12 + H1∗ Z2 cos ξ, (6.8.27) sin ξ sin η1 sin η2

Z1 = − (1 + b∗ H1∗ ) Z1 Z4

cos ξ cos ξ cos η1 + (1 + b∗ H1∗ ) Z1 Z3 sin ξ sin ξ sin η1 cos ξ 1 cos η2 − (1 + b∗ H1∗ ) Z1 Z2 + H1∗ Z1 cos ξ, (6.8.28) sin ξ sin η1 sin η2 cos ξ , sin ξ cos ξ , η2 = (1 + b∗ H1∗ ) Z2 sin ξ sin η1 cos ξ , η3 = − (1 + b∗ H1∗ ) Z1 sin ξ sin η1 sin η2

η1 = − (1 + b∗ H1∗ ) Z3

(6.8.29) (6.8.30) (6.8.31)

on the tangent bundle T∗ S4 {(Z4 , Z3 , Z2 , Z1 ; ξ, η1 , η2 , η3 ) ∈ R8 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}

(6.8.32)

of the four-dimensional sphere S4 {(ξ, η1 , η2 , η3 ) ∈ R4 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. 371

We see that the independent eighth-order subsystem (6.8.24)–(6.8.31) (due to the fact that the variable η3 is cyclic) can be substituted into the seventh-order system (6.8.24)–(6.8.30) and can be considered separately on its own seven-dimensional manifold. 6.8.3. Complete list of the first integrals for any finite n. We turn now to the integration of the desired system (6.8.13)–(6.8.22) of the order 2(n − 1) (without any simplifications, i.e., in the presence of all coefficients). Similarly, to integrate completely the system (6.8.13)–(6.8.22) of the order 2(n − 1), in general, we need 2n − 3 independent first integrals. However, after the change of variables ⎞ ⎛ ⎞ ⎛ wn−1 Zn−1 ⎜ wn−2 ⎟ ⎜ Zn−2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ... ⎟ → ⎜ ... ⎟, ⎟ ⎜ ⎟ ⎜ ⎝ w2 ⎠ ⎝ Z2 ⎠ Z1 w1 wn−1 = −Zn−1 ,

$ wn−2 =

2 , Z12 + . . . + Zn−2

Zn−3 , w2 = − $ 2 2 Z1 + . . . + Zn−4

wn−3 =

Z2 , Z1

wn−4 = − (

Z3 Z12 + Z22

,

. . . , (6.8.33)

Zn−2 w1 = − $ , 2 2 Z1 + . . . + Zn−3

the system (6.8.13)–(6.8.22) splits as follows: ⎧  ξ = −(1 + b∗ H1∗ )wn−1 − b∗ sin ξ, ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 + H1∗ wn−1 cos ξ, wn−1 = sin ξ cos ξ − (1 + b∗ H1∗ )wn−2 sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ + H1∗ wn−2 cos ξ, = (1 + b∗ H1∗ )wn−2 wn−1 ⎩wn−2 sin ξ ⎧ ⎪ 1 + ws2 cos ηs ⎨w = d (w , s n−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) s ws sin ηs ⎪ ⎩η  = d (w s = 1, . . . , n − 3, s n−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ), s  ηn−2 = dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ),

(6.8.34)

(6.8.35) (6.8.36)

where cos ξ , sin ξ cos ξ , d2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = (1 + b∗ H1∗ )Zn−3 (wn−1 , . . . , w1 ) sin ξ sin η1 ............................................................ d1 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = −(1 + b∗ H1∗ )Zn−2 (wn−1 , . . . , w1 )

dn−2 (wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) = (−1)n (1 + b∗ H1∗ ) × Z1 (wn−1 , . . . , w1 )

cos ξ ; sin ξ sin η1 . . . sin ηn−3

in this case Zk = Zk (wn−1 , . . . , w1 ), are the functions by virtue of change (6.8.33). 372

k = 1, . . . , n − 2,

(6.8.37)

In particular, for n = 5 we obtain the following eighth-order system: ⎧  ξ = −(1 + b∗ H1∗ )w4 − b∗ sin ξ, ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  + H1∗ w4 cos ξ, w4 = sin ξ cos ξ − (1 + b∗ H1∗ )w32 sin ξ ⎪ ⎪ ⎪ cos ξ ⎪ ⎪ + H1∗ w3 cos ξ, ⎩w3 = (1 + b∗ H1∗ )w3 w4 sin ξ ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w2 cos η2 , 2 4 3 2 1 1 2 3 2 w2 sin η2 ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; ξ, η , η , η ) 1 + w1 cos η1 , 1 4 3 2 1 1 2 3 1 w1 sin η1 ⎪ ⎩η  = d (w , w , w , w ; ξ, η , η , η ), 1 4 3 2 1 1 2 3 1 η3 = d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ),

(6.8.38)

(6.8.39)

(6.8.40) (6.8.41)

where d1 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = −Z3 (w4 , w3 , w2 , w1 )

cos ξ w1 w3 cos ξ = ∓( , sin ξ 1 + w12 sin ξ

cos ξ sin ξ sin η1 w2 w3 cos ξ ( = ±( , 1 + w12 1 + w22 sin ξ sin η1 cos ξ d3 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = −Z1 (w4 , w3 , w2 , w1 ) sin ξ sin η1 sin η2 w3 cos ξ ( = ∓( , 2 2 sin ξ sin η1 sin η2 1 + w1 1 + w 2 d2 (w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) = Z2 (w4 , w3 , w2 , w1 )

(6.8.42)

in this case Zk = Zk (w4 , w3 , w2 , w1 ),

k = 1, 2, 3,

(6.8.43)

are the functions by virtue of change (6.8.33). The system (6.8.34)–(6.8.36) is studied on the tangent bundle T∗ Sn−1 {(wn−1 , . . . , w1 ; ξ, η1 , . . . , ηn−2 ) ∈ R2(n−1) : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π} (6.8.44) of the (n − 1)-dimensional sphere Sn−1 {(ξ, η1 , . . . , ηn−2 ) ∈ Rn−1 : 0 ≤ ξ, η1 , . . . , ηn−3 ≤ π, ηn−2 mod 2π}. In particular, the system (6.8.38)–(6.8.41) is studied on the tangent bundle T∗ S4 {(w4 , w3 , w2 , w1 ; ξ, η1 , η2 , η3 ) ∈ R8 : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π} S4 {(ξ, η1 , η2 , η3 )

(6.8.45)

R4

∈ : 0 ≤ ξ, η1 , η2 ≤ π, η3 mod 2π}. of the four-dimensional sphere We see that the independent subsystem (6.8.34) (which can be considered separately on its own three-dimensional manifold) and n − 3 independent second-order subsystems (4.7.21) (after the change of independent variable) can be substituted into the system (6.8.34)–(6.8.36) of the order 2(n − 1), and also Eq. (6.8.36) for ηn−2 is separated (due to the fact that the variable ηn−2 is cyclic). In particular, we see that the independent third-order subsystem (6.8.38) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystems (6.8.39) and (6.8.40) (after the change of independent variable) can be substituted into the 373

eighth-order system (6.8.38)–(6.8.40), and also Eq. (6.8.41) for η3 is separated (due to the fact that the variable η3 is cyclic). Thus, to integrate completely the system (6.8.34)–(6.8.36), it suffices to specify two independent first integrals of system (6.8.34), one by one the first integral of systems (6.8.35) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.8.36) (i.e., only n). In particular, to integrate completely the system (6.8.38)–(6.8.41), it suffices to specify two independent first integrals of system (6.8.38), one by one the first integral of systems (6.8.39), (6.8.40), and an additional first integral that “attaches” Eq. (6.8.41) (i.e., only five). First, we compare the third-order system (6.8.34) with the nonautonomous second-order system ⎧ 2 sin ξ cos ξ − (1 + b∗ H1∗ )wn−2 cos ξ/ sin ξ + H1∗ wn−1 cos ξ ⎪ dwn−1 ⎪ ⎪ = , ⎨ dξ −(1 + b∗ H1∗ )wn−1 − b∗ sin ξ (6.8.46) ⎪ dwn−2 (1 + b∗ H1∗ )wn−2 wn−1 cos ξ/ sin ξ + H1∗ wn−2 cos ξ ⎪ ⎪ = . ⎩ dξ −(1 + b∗ H1∗ )wn−1 − b∗ sin ξ Using the substitution τ = sin ξ, we rewrite system (6.8.46) in the algebraic form: ⎧ 2 τ − (1 + b∗ H1∗ )wn−2 /τ + H1∗ wn−1 ⎪ dwn−1 ⎪ ⎪ = , ⎨ dτ −(1 + b∗ H1∗ )wn−1 − b∗ τ ⎪ (1 + b∗ H1∗ )wn−2 wn−1 /τ + H1∗ wn−2 dwn−2 ⎪ ⎪ = . ⎩ dτ −(1 + b∗ H1∗ )wn−1 − b∗ τ

(6.8.47)

Further, if we introduce the uniform variables by the formulas wn−1 = u2 τ,

wn−2 = u1 τ,

(6.8.48)

we reduce system (6.8.47) to the following form: ⎧ 1 − (1 + b∗ H1∗ )u21 + H1∗ u2 du2 ⎪ ⎪ + u = , τ 2 ⎨ dτ −(1 + b H )u − b ∗

1∗

2

∗

⎪ (1 + b∗ H1∗ )u1 u2 + H1∗ u1 du ⎪ ⎩τ 1 + u1 = , dτ −(1 + b∗ H1∗ )u2 − b∗

(6.8.49)

which is equivalent to ⎧ (1 + b∗ H1∗ )(u22 − u21 ) + (b∗ + H1∗ )u2 + 1 du2 ⎪ ⎪ = , τ ⎨ dτ −(1 + b∗ H1∗ )u2 − b∗ ⎪ 2(1 + b∗ H1∗ )u1 u2 + (b∗ + H1∗ )u1 du ⎪ ⎩τ 1 = . dτ −(1 + b∗ H1∗ )u2 − b∗

(6.8.50)

We compare the second-order system (6.8.50) with the nonautonomous first-order equation 1 − (1 + b∗ H1∗ )(u21 − u22 ) + (b∗ + H1∗ )u2 du2 = , du1 2(1 + b∗ H1∗ )u1 u2 + (b∗ + H1∗ )u1 which can be easily reduced to the exact differential equation  (1 + b∗ H1∗ )(u22 + u21 ) + (b∗ + H1∗ )u2 + 1 = 0. d u1

(6.8.51)

(6.8.52)

Therefore, Eq. (6.8.51) has the following first integral: (1 + b∗ H1∗ )(u22 + u21 ) + (b∗ + H1∗ )u2 + 1 = C1 = const, u1 374

(6.8.53)

which in the old variables has the form 2 2 (1 + b∗ H1∗ )(wn−1 + wn−2 ) + (b∗ + H1∗ )wn−1 sin ξ + sin2 ξ Θ1 (wn−1 , wn−2 ; ξ) = = C1 = const . wn−2 sin ξ (6.8.54) Remark 6.8.1. We consider system (6.8.34) with variable dissipation with zero mean (see [161, 168, 199]), which becomes conservative for b∗ = H1∗ : ⎧  ξ = −(1 + b2∗ )wn−1 − b∗ sin ξ, ⎪ ⎪ ⎪ ⎪ ⎪ cos ξ ⎨  2 + b∗ wn−1 cos ξ, wn−1 = sin ξ cos ξ − (1 + b2∗ )wn−2 (6.8.55) sin ξ ⎪ ⎪ ⎪ cos ξ ⎪  ⎪ ⎩wn−2 = (1 + b2∗ )wn−2 wn−1 + b∗ wn−2 cos ξ. sin ξ It has two analytical first integrals of the form 2 2 + wn−2 ) + 2b∗ wn−1 sin ξ + sin2 ξ = C1∗ = const, (1 + b2∗ )(wn−1

(6.8.56)

wn−2 sin ξ = C2∗ = const .

(6.8.57)

It is obvious that the ratio of the first integrals (6.8.56), (6.8.57) is also a first integral of the system (6.8.55). However, for b∗ = H1∗ both functions 2 2 (1 + b∗ H1∗ )(wn−1 + wn−2 ) + (b∗ + H1∗ )wn−1 sin ξ + sin2 ξ

(6.8.58)

and (6.8.57) are not first integrals of system (6.8.34), but their ratio (i.e., the ratio of the functions (6.8.58) and (6.8.57)) is a first integral of system (6.8.34) for any b∗ , H1∗ . In the sequel, we find the obvious form of the additional first integral of the third-order system (6.8.34). For this, we first transform the invariant relation (6.8.53) for u1 = 0 as follows: 2  2  b∗ + H1∗ C1 (b∗ − H1∗ )2 + C12 − 4 + u1 − = . (6.8.59) u2 + 2(1 + b∗ H1∗ ) 2(1 + b∗ H1∗ ) 4(1 + b∗ H1∗ )2 We see that the parameters of the given invariant relation must satisfy the condition (b∗ − H1∗ )2 + C12 − 4 ≥ 0,

(6.8.60)

and the phase space of system (6.8.34) is stratified into a family of surfaces defined by Eq. (6.8.59). Thus, by virtue of relation (6.8.53) the first equation of system (6.8.50) has the form τ

2(1 + b∗ H1∗ )u22 + 2(b∗ + H1∗ )u2 + 2 − C1 U1 (C1 , u2 ) du2 = , dτ −b∗ − (1 + b∗ H1∗ )u2

where U1 (C1 , u2 ) = $ U2 (C1 , u2 ) =

1 {C1 ± U2 (C1 , u2 )}, 2(1 + b∗ H1∗ )

(6.8.61)

(6.8.62)

C12 − 4(1 + b∗ H1∗ )(1 + (b∗ + H1∗ )u2 + (1 + b∗ H1∗ )u22 ),

and the integration constant C1 is chosen from condition (6.8.60). Therefore, the quadrature for the search of an additional first integral of system (6.8.34) has the form & & (−b∗ − (1 + b∗ H1∗ )u2 )du2 dτ = . (6.8.63) τ 2(1 + (b∗ + H1∗ )u2 + (1 + b∗ H1∗ )u22 ) − C1 {C1 ± U2 (C1 , u2 )}/(2(1 + b∗ H1∗ )) Obviously, the left-hand side up to an additive constant is equal to ln | sin ξ|.

(6.8.64) 375

If b∗ + H1∗ = r1 , b21 = (b∗ − H1∗ )2 + C12 − 4, (6.8.65) 2(1 + b∗ H1∗ ) then the right-hand side of Eq. (6.8.63) has the form & 1 d(b21 − 4(1 + b∗ H1∗ )r12 ) ( − 4 (b21 − 4(1 + b∗ H1∗ )r12 ) ± C1 b21 − 4(1 + b∗ H1∗ )r12 & dr1 ( + (b∗ − H1∗ )(1 + b∗ H1∗ ) 2 2 (b1 − 4(1 + b∗ H1∗ )r1 ) ± C1 b21 − 4(1 + b∗ H1∗ )r12 (   −b + H 1  b21 − 4(1 + b∗ H1∗ )r12  ∗ 1∗ I1 , (6.8.66) = − ln  ± 1 ±  2  C1 2 u2 +

where

& I1 =

(

$

dr3 b21 − r32 (r3 ± C1 )

,

r3 =

b21 − 4(1 + b∗ H1∗ )r12 .

(6.8.67)

In the calculation of integral (6.8.67), the following three cases are possible. I. |b∗ − H1∗ | > 2. (   (b − H )2 − 4 + (b2 − r 2  1 C   ∗ 1∗ 1 1 3 ln  ±( I1 = − (  2 2 r3 ± C 1 2 (b∗ − H1∗ ) − 4  (b∗ − H1∗ ) − 4  (   (b − H )2 − 4 − (b2 − r 2  C 1   ∗ 1∗ 1 1 3 ln  ∓( + (  + const . (6.8.68) r3 ± C1 2 (b∗ − H1∗ )2 − 4  (b∗ − H1∗ )2 − 4  II. |b∗ − H1∗ | < 2. I1 = (

1 4 − (b∗ − H1∗ )2

III. |b∗ − H1∗ | = 2.

arcsin

±C1 r3 + b21 + const . b1 (r3 ± C1 )

(6.8.69)

(

b21 − r32 + const . C1 (r3 ± C1 )

(6.8.70)

wn−1 b∗ + H1∗ + , sin ξ 2(1 + b∗ H1∗ )

(6.8.71)

I1 = ∓ When we return to the variable r1 =

we obtain the final form for the value I1 : I. |b∗ − H1∗ | > 2. (   (b − H )2 − 4 ± 2(1 + b H )r  1 C1   ∗ 1∗ ∗ 1∗ 1 ( ( ln  ± I1 = − (  2 2 2 2 2  2 (b∗ − H1∗ ) − 4 (b∗ − H1∗ ) − 4  b1 − 4(1 + b∗ H1∗ ) r1 ± C1 (   (b − H )2 − 4 ∓ 2(1 + b H )r  C1 1   ∗ 1∗ ∗ 1∗ 1 ( ( ln  ∓ + (  + const . 2 2 2 2 2  2 (b∗ − H1∗ ) − 4 (b∗ − H1∗ ) − 4  b1 − 4(1 + b∗ H1∗ ) r1 ± C1 (6.8.72) II. |b∗ − H1∗ | < 2.

( ±C1 b21 − 4(1 + b∗ H1∗ )2 r12 + b21 ( + const . arcsin I1 = ( 4 − (b∗ − H1∗ )2 b1 ( b21 − 4(1 + b∗ H1∗ )2 r12 ± C1 ) 1

376

(6.8.73)

III. |b∗ − H1∗ | = 2. I1 = ∓

2(1 + b∗ H1∗ )r1 ( + const . C1 ( b21 − 4(1 + b∗ H1∗ )2 r12 ± C1 )

(6.8.74)

Thus, we have found an additional first integral for the third-order system (6.8.34), i.e., we have a complete set of first integrals that are transcendental functions of the phase variables. Remark 6.8.2. In the expression of the found first integral, we must formally substitute the left-hand side of the first integral (6.8.53) instead of C1 . Then the obtained additional first integral has the following structure:  wn−1 wn−2 , = C2 = const . Θ2 (wn−1 , wn−2 ; ξ) = G sin ξ, sin ξ sin ξ

(6.8.75)

Thus, we have found two first integrals (6.8.54), (6.8.75) of the independent third-order system (6.8.34). To integrate it completely, it suffices to find one by one the first integral for the systems (6.8.35) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.8.36). Indeed, the desired first integrals coincide with the first integrals (6.7.371) and (6.7.372), namely, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, (6.8.76) Θs+2 (ws ; ηs ) = sin ηs Cn−1 cos ηn−3 = Cn = const; (6.8.77) Θn (wn−3 , wn−4 ; ηn−4 , ηn−3 , ηn−2 ) = ηn−2 ± arctg $ 2 2 Cn−2 sin2 ηn−3 − Cn−1 in this case, on the left-hand side of Eq. (6.8.77), we must substitute instead of Cn−2 , Cn−1 the first integrals (6.8.76) for s = n − 4, n − 3. Theorem 6.8.2. The system (6.8.34)–(6.8.36) of order 2(n − 1) possesses a sufficient number (n) of independent first integrals (6.8.54), (6.8.75), (6.8.76), and (6.8.77). Therefore, in the considered case, the system of dynamical equations (6.8.34)–(6.8.36) has n first integrals expressed by the relations (6.8.54), (6.8.75), (6.8.76), and (6.8.77), which are transcendental functions of the phase variables (in the sense of the complex analysis) and are expressed as a finite combination of elementary functions (in this case, we use the expressions (6.8.63)–(6.8.74)). Theorem 6.8.3. Three sets of the relations (6.3.1), (6.4.3), and (6.5.9) under the conditions (6.3.6)– (6.3.8), (6.8.4), and (6.8.8) possess n first integrals (a complete set), which are transcendental functions (in the sense of complex analysis) and are expressed as a finite combination of elementary functions. 6.8.4. Topological analogies. Now we present two groups of analogies related to the system (6.6.1), which describes the motion of a free body in the presence of a tracking force. The first group of analogies deals with the case of the presence of the nonintegrable constraint (6.6.12) in the system. In this case the dynamical part of the motion equations under certain conditions is reduced to the system (6.6.31). Under the conditions (6.8.4) and (6.8.8), the system (6.6.31) has the form α = − (1 + bH1 ) Zn−1 + b sin α,  Zn−1

(6.8.78)

cos α 2 − H1 Zn−1 cos α, = sin α cos α − (1 + bH1 ) (Z12 + . . . + Zn−2 ) sin α

 = (1 + bH1 ) Zn−2 Zn−1 Zn−2

(6.8.79)

cos α sin α 2 ) + (1 + bH1 ) (Z12 + . . . + Zn−3

cos α cos β1 − H1 Zn−2 cos α, (6.8.80) sin α sin β1 377

cos α cos α cos β1 − (1 + bH1 ) Zn−3 Zn−2 sin α sin α sin β1 cos α 1 cos β2 2 − (1 + bH1 ) (Z12 + . . . + Zn−4 ) − H1 Zn−3 cos α, (6.8.81) sin α sin β1 sin β2 .......................................... n−2   cos α cos β s−1 (−1)s+1 Zn−s − H1 Z1 cos α, (6.8.82) Z1 = (1 + bH1 ) Z1 sin α s=1 sin β1 . . . sin βs−1

 Zn−3 = (1 + bH1 ) Zn−3 Zn−1

cos α , sin α cos α , β2 = − (1 + bH1 ) Zn−3 sin α sin β1 .......................................... cos α  βn−3 = (−1)n (1 + bH1 ) Z2 , sin α sin β1 . . . sin βn−4 cos α  βn−2 = (−1)n+1 (1 + bH1 ) Z1 . sin α sin β1 . . . sin βn−3 β1 = (1 + bH1 ) Zn−2

(6.8.83) (6.8.84)

(6.8.85) (6.8.86)

If we introduce the dimensionless parameters, the variables, and differentiation analogously to (6.8.9): b = σn0 ,

AB h1 B , H1 = , (n − 2)I2 (n − 2)I2 n0   k = 1, . . . , n − 1, · = n0 v .

n20 =

zk = n0 vZk ,

(6.8.87)

In particular, for n = 5 we obtain the following eighth-order system: α = − (1 + bH1 ) Z4 + b sin α, cos α − H1 Z4 cos α, sin α cos α cos β1 cos α + (1 + bH1 ) (Z12 + Z22 ) Z3 = (1 + bH1 ) Z3 Z4 − H1 Z3 cos α, sin α sin α sin β1 Z4 = sin α cos α − (1 + bH1 ) (Z12 + Z22 + Z32 )

(6.8.88) (6.8.89) (6.8.90)

Z2 = (1 + bH1 ) Z2 Z4

cos α cos α cos β1 − (1 + bH1 ) Z2 Z3 sin α sin α sin β1 cos α 1 cos β2 − (1 + bH1 ) Z12 − H1 Z2 cos α, (6.8.91) sin α sin β1 sin β2

Z1 = (1 + bH1 ) Z1 Z4

cos α cos α cos β1 − (1 + bH1 ) Z1 Z3 sin α sin α sin β1 cos α 1 cos β2 + (1 + bH1 ) Z1 Z2 − H1 Z1 cos α, (6.8.92) sin α sin β1 sin β2 cos α , sin α cos α β2 = − (1 + bH1 ) Z2 , sin α sin β1 cos α . β3 = (1 + bH1 ) Z1 sin α sin β1 sin β2

β1 = (1 + bH1 ) Z3

(6.8.93) (6.8.94) (6.8.95)

Theorem 6.8.4. System (6.8.78)–(6.8.86) (for the case of a free body) is equivalent to the system (6.8.13)–(6.8.22) (for the case of a fixed pendulum). 378

Indeed, it is sufficient to substitute ξ = α,

η1 = β1 ,

...,

ηn−2 = βn−2 ,

b∗ = −b,

H1∗ = −H1 ,

(6.8.96)

and also to compare the variables Zk ↔ −Zk , k = 1, . . . , n − 1. To integrate completely the system (6.8.78)–(6.8.86) of the order 2(n − 1), in general, we need 2n − 3 independent first integrals (in particular, to integrate completely the eighth-order system (6.8.88)–(6.8.95), in general, we need seven independent first integrals). However, after the change of variables ⎞ ⎛ ⎞ ⎛ wn−1 Zn−1 ⎜ Zn−2 ⎟ ⎜ wn−2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ... ⎟ → ⎜ ... ⎟, ⎟ ⎜ ⎟ ⎜ ⎝ Z2 ⎠ ⎝ w2 ⎠ Z1 w1 $ Z2 Z3 2 , wn−3 = , wn−4 = ( 2 , . . . , (6.8.97) wn−1 = Zn−1 , wn−2 = Z12 + . . . + Zn−2 Z1 Z1 + Z22 w2 = $

Zn−3 2 Z12 + . . . + Zn−4

, w1 = $

Zn−2 2 Z12 + . . . + Zn−3

,

the system (6.8.78)–(6.8.86) splits as follows: ⎧  α = −(1 + bH1 )wn−1 + b sin α, ⎪ ⎪ ⎪ ⎨  cos α 2 − H1 wn−1 cos α, wn−1 = sin α cos α − (1 + bH1 )wn−2 sin α ⎪ ⎪ cos α ⎪ ⎩w − H1 wn−2 cos α, n−2 = (1 + bH1 )wn−2 wn−1 sin α ⎧ ⎪ 1 + ws2 cos βs ⎨w = d (w , s n−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) s ws sin βs ⎪ ⎩β  = d (w s = 1, . . . , n − 3, s n−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), s

(6.8.98)

(6.8.99)

 βn−2 = dn−2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ),

(6.8.100)

cos α , sin α cos α , d2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = −(1 + bH1 )Zn−3 (wn−1 , . . . , w1 ) sin α sin β1 ............................................................

(6.8.101)

where d1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = (1 + bH1 )Zn−2 (wn−1 , . . . , w1 )

dn−2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = (−1)n+1 (1 + bH1 ) × Z1 (wn−1 , . . . , w1 )

cos α ; sin α sin β1 . . . sin βn−3

in this case Zk = Zk (wn−1 , . . . , w1 ),

k = 1, . . . , n − 2,

are the functions, by virtue of change (6.8.97). In particular, for n = 5 the system (6.8.88)–(6.8.95) splits as follows: ⎧  α = −(1 + bH1 )w4 + b sin α, ⎪ ⎪ ⎪ ⎨ cos α − H1 w4 cos α, w4 = sin α cos α − (1 + bH1 )w32 sin α ⎪ ⎪ ⎪ ⎩w = (1 + bH1 )w3 w4 cos α − H1 w3 cos α, 3 sin α

(6.8.102)

(6.8.103)

379

⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w2 cos β2 , 2 4 3 2 1 1 2 3 2 w2 sin β2 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w1 cos β1 , 1 4 3 2 1 1 2 3 1 w1 sin β1 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 1 4 3 2 1 1 2 3 1 β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(6.8.104)

(6.8.105) (6.8.106)

where cos α sin α w1 w3 cos α = ±(1 + bH1 ) ( , 1 + w12 sin α

d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = (1 + bH1 )Z3 (w4 , w3 , w2 , w1 )

cos α sin α sin β1 w2 w3 cos α ( = ∓(1 + bH1 ) ( , (6.8.107) 2 2 1 + w1 1 + w2 sin α sin β1

d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = −(1 + bH1 )Z2 (w4 , w3 , w2 , w1 )

cos α sin α sin β1 sin β2 w3 cos α ( = ±(1 + bH1 ) ( ; 2 2 1 + w1 1 + w2 sin α sin β1 sin β2

d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = (1 + bH1 )Z1 (w4 , w3 , w2 , w1 )

in this case Zk = Zk (w4 , w3 , w2 , w1 ), k = 1, 2, 3, are the functions, by virtue of change (6.8.97). The system (6.8.98)–(6.8.100) is studied on the tangent bundle

(6.8.108)

T∗ Sn−1 {(wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) ∈ R2(n−1) : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π} (6.8.109) of the (n − 1)-dimensional sphere Sn−1 {(α, β1 , . . . , βn−2 ) ∈ Rn−1 : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π}. In particular, the system (6.8.103)–(6.8.106) is studied on the tangent bundle T∗ S4 {(w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) ∈ R8 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}

(6.8.110)

of the four-dimensional sphere S4 {(α, β1 , β2 , β3 ) ∈ R4 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}. We see that the independent third-order subsystem (6.8.98) (which can be considered separately on its own three-dimensional manifold) and n − 3 independent second-order subsystems (6.8.99) (after the change of independent variable) can be substituted into the system (6.8.98)–(6.8.100) of the order 2(n − 1), and also Eq. (6.8.100) on βn−2 is separated (due to the fact that the variable βn−2 is cyclic). In particular, we see that the independent third-order subsystem (6.8.103) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystem (6.8.104) and (6.8.105) (after the change of independent variable) can be substituted into the eighth-order system (6.8.103)–(6.8.106), and also Eq. (6.8.106) on β3 is separated (due to the fact that the variable β3 is cyclic). Thus, to integrate completely the system (6.8.98)–(6.8.100), it suffices to specify two independent first integrals of system (6.8.98), one by one the first integral of systems (6.8.99) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.8.100) (i.e., only n). 380

In particular, to integrate completely the system (6.8.103)–(6.8.106), it suffices to specify two independent first integrals of system (6.8.103), one by one the first integral of systems (6.8.104), (6.8.105), and an additional first integral that “attaches” Eq. (6.8.106) (i.e., only five). Corollary 6.8.1. 1. The attack angle α and the angles β1 , . . . , βn−2 in the case of a free body are equivalent to the angles of the body’s deviation ξ and η1 , . . . , ηn−2 for a fixed pendulum. 2. The distance σ = CD for a free body corresponds to the length of a holder l = OD of a fixed pendulum. 3. The first integrals of the system (6.8.98)–(6.8.100) can be automatically obtained through Eqs. (6.8.54), (6.8.75), (6.8.76), and (6.8.77) after the substitutions (6.8.96) (see also [199]): Θ1 (wn−1 , wn−2 ; α) =

2 2 (1 + bH1 )(wn−1 + wn−2 ) − (b + H1 )wn−1 sin α + sin2 α = C1 = const . wn−2 sin α (6.8.111)

 wn−1 wn−2  , = C2 = const . Θ2 (wn−1 , wn−2 ; α) = G sin α, sin α sin α

(6.8.112)

( Θs+2 (ws ; βs )

=

1 + ws2 = Cs+2 = const, sin βs

Θn (wn−3 , wn−4 ; βn−4 , βn−3 , βn−2 ) = βn−2 ± arctg $

s = 1, . . . , n − 3,

Cn−1 cos βn−3 2 2 Cn−2 sin2 βn−3 − Cn−1

(6.8.113)

= Cn = const; (6.8.114)

in this case, on the left-hand side of Eq. (6.8.114), we must substitute instead of Cn−2 , Cn−1 the first integrals (6.8.113) for s = n − 4, n − 3. The second group of analogies deals with the case of the motion with constant velocity of the center of mass of a body, i.e., when the property (6.6.39) holds. In this case the dynamical part of the motion equations under certain conditions is reduced to the system (6.6.45)–(6.6.54). Then, under the conditions (6.6.39), (6.8.4), (6.8.8), and (6.8.87), the reduced dynamical part of the motion equations (system (6.6.46)–(6.6.54)) has the form of the analytical system n−1    Zs2 sin α + b sin α cos2 α − bH1 Zn−1 cos2 α, (6.8.115) α = −Zn−1 + b s=1

 = sin α cos α − (1 + bH1 ) Zn−1

n−2  s=1

 Zs2 2

cos α + bZn−1 sin α

− bZn−1 sin α cos α +

 Zn−2

n−1 

 Zs2 cos α

s=1 2 bH1 Zn−1 sin α cos α −

H1 Zn−1 cos α, (6.8.116)

n−3  n−1    cos α cos α cos β1 2 = (1 + bH1 )Zn−2 Zn−1 + (1 + bH1 ) Zs + bZn−2 Zs2 cos α sin α sin α sin β 1 s=1 s=1 − bZn−2 sin2 α cos α + bH1 Zn−2 Zn−1 sin α cos α − H1 Zn−2 cos α, (6.8.117)

381

cos α cos α cos β1  Zn−3 = (1 + bH1 )Zn−3 Zn−1 − (1 + bH1 )Zn−3 Zn−2 sin α sin α sin β1 n−4  n−1    cos α 1 cos β 2 − (1 + bH1 ) Zs2 + bZn−3 Zs2 cos α sin α sin β sin β 1 2 s=1 s=1 − bZn−3 sin2 α cos α + bH1 Zn−3 Zn−1 sin α cos α − H1 Zn−3 cos α, (6.8.118) ................................................ Z1

cos α = (1 + bH1 )Z1 sin α

n−2  (−1)s+1 Zn−s s=1

cos βs−1 sin β1 . . . sin βs−1

 + bZ1

n−1  s=1

 Zs2

cos α

− bZ1 sin2 α cos α + bH1 Z1 Zn−1 sin α cos α − H1 Z1 cos α, (6.8.119) cos α , sin α cos α , β2 = −(1 + bH1 )Zn−3 sin α sin β1 .......................................... cos α  = (−1)n (1 + bH1 )Z2 , βn−3 sin α sin β1 . . . sin βn−4 cos α  = (−1)n+1 (1 + bH1 )Z1 ; βn−2 sin α sin β1 . . . sin βn−3 β1 = (1 + bH1 )Zn−2

(6.8.120) (6.8.121)

(6.8.122) (6.8.123)

in this case, we choose the constant n1 as follows: n1 = n0 .

(6.8.124)

In particular, for n = 5 we obtain the following eighth-order system: α = −Z3 + b(Z12 + Z22 + Z32 + Z42 ) sin α + b sin α cos2 α − bH1 Z4 cos2 α,

(6.8.125)

cos α + bZ4 (Z12 + Z22 + Z32 + Z42 ) cos α sin α − bZ4 sin2 α cos α + bH1 Z42 sin α cos α − H1 Z4 cos α, (6.8.126)

Z4 = sin α cos α − (1 + bH1 )(Z12 + Z22 + Z32 )

Z3 = (1 + bH1 )Z3 Z4

cos α cos α cos β1 + (1 + bH1 )(Z12 + Z22 ) + bZ3 (Z12 + Z22 + Z32 + Z42 ) cos α sin α sin α sin β1 − bZ3 sin2 α cos α + bH1 Z3 Z4 sin α cos α − H1 Z3 cos α, (6.8.127)

cos α cos α cos β1 cos α 1 cos β2 − (1 + bH1 )Z2 Z3 − (1 + bH1 )Z12 sin α sin α sin β1 sin α sin α sin β2 2 2 2 2 2 + bZ2 (Z1 + Z2 + Z3 + Z4 ) cos α − bZ2 sin α cos α + bH1 Z2 Z4 sin α cos α − H1 Z2 cos α, (6.8.128) Z2 = (1 + bH1 )Z2 Z4

cos α cos α cos β1 cos α 1 cos β2 − (1 + bH1 )Z1 Z3 + (1 + bH1 )Z1 Z2 sin α sin α sin β1 sin α sin α sin β2 2 2 2 2 2 + bZ1 (Z1 + Z2 + Z3 + Z4 ) cos α − bZ1 sin α cos α + bH1 Z1 Z4 sin α cos α − H1 Z1 cos α, (6.8.129) Z1 = (1 + bH1 )Z1 Z4

382

cos α , sin α cos α , β2 = −(1 + bH1 )Z2 sin α sin β1 cos α β3 = (1 + bH1 )Z1 . sin α sin β1 sin β2 β1 = (1 + bH1 )Z3

(6.8.130) (6.8.131) (6.8.132)

To integrate completely the system (6.8.115)–(6.8.123) of order 2(n − 1), in general, we need 2n − 3 independent first integrals. However, after the change of variables (6.8.97) the system (6.8.115)– (6.8.123) splits as follows: ⎧  2 2 + wn−1 ) sin α + b sin α cos2 α − bH1 wn−1 cos2 α, α = −wn−1 + b(wn−2 ⎪ ⎪ ⎪ ⎪ cos α ⎪  2 2 2 ⎪ ⎪ + bwn−1 (wn−2 wn−1 = sin α cos α − (1 + bH1 )wn−2 + wn−1 ) cos α ⎪ ⎪ sin α ⎨ 2 sin α cos α − H1 wn−1 cos α, − bwn−1 sin2 α cos α + bH1 wn−1 (6.8.133) ⎪ ⎪ cos α ⎪  2 2 ⎪ ⎪ wn−2 = (1 + bH1 )wn−2 wn−1 + bwn−2 (wn−2 + wn−1 ) cos α ⎪ ⎪ sin α ⎪ ⎪ ⎩ − bwn−2 sin2 α cos α + bH1 wn−2 wn−1 sin α cos α − H1 wn−2 cos α, ⎧ s ⎨w = d1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) 1 + w1 cos βs , s ws sin βs (6.8.134) ⎩  βs = d1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), s = 1, . . . , n − 3,  βn−2 = d2 (wn−1 , . . . , w1 ; α, β1 , . . . , β2 ),

where the conditions (6.8.101) hold. In particular, for n = 5 the system (6.8.125)–(6.8.132) splits as follows: ⎧  2 2 2 2 ⎪ ⎪α = −w4 + b(w3 + w4 ) sin α + b sin α cos α − bH1 w4 cos α, ⎪ ⎪ cos α ⎪ ⎪ ⎪w4 = sin α cos α − (1 + bH1 )w32 + bw4 (w32 + w42 ) cos α ⎪ ⎪ sin α ⎨ − bw4 sin2 α cos α + bH1 w42 sin α cos α − H1 w4 cos α, ⎪ ⎪ cos α ⎪  ⎪ ⎪ + bw3 (w32 + w42 ) cos α w3 = (1 + bH1 )w3 w4 ⎪ ⎪ sin α ⎪ ⎪ ⎩ − bw3 sin2 α cos α + bH1 w3 wn−1 sin α cos α − H1 w3 cos α, ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w2 cos β2 , 2 4 3 2 1 1 2 3 2 w2 sin β2 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 2 4 3 2 1 1 2 3 2 ⎧ 2 ⎪ ⎨w = d (w , w , w , w ; α, β , β , β ) 1 + w1 cos β1 , 1 4 3 2 1 1 2 3 1 w1 sin β1 ⎪ ⎩β  = d (w , w , w , w ; α, β , β , β ), 1 4 3 2 1 1 2 3 1 β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(6.8.135)

(6.8.136)

(6.8.137)

(6.8.138) (6.8.139)

where the conditions (6.8.107) hold. The system (6.8.133)–(6.8.135) is studied on the tangent bundle (6.8.109) of the (n − 1)-dimensional sphere Sn−1 {(α, β1 , . . . , βn−2 ) ∈ Rn−1 : 0 ≤ α, β1 , . . . , βn−3 ≤ π, βn−2 mod 2π}. In particular, the system (6.8.136)–(6.8.139) is studied on the tangent bundle (6.8.110) of the fourdimensional sphere S4 {(α, β1 , β2 , β3 ) ∈ R4 : 0 ≤ α, β1 , β2 ≤ π, β3 mod 2π}. We see that the independent third-order subsystem (6.8.133) (which can be considered separately on its own three-dimensional manifold) and n−3 independent second-order subsystems (6.8.134) (after 383

the change of independent variable) can be substituted into the system (6.8.133)–(6.8.135) of the order 2(n − 1), and also Eq. (6.8.135) on βn−2 is separated (due to the fact that the variable βn−2 is cyclic). In particular, we see that the independent third-order subsystem (6.8.136) (which can be considered separately on its own three-dimensional manifold) and two independent second-order subsystem (6.8.137) and (6.8.138) (after the change of independent variable) can be substituted into the eighth-order system (6.8.136)–(6.8.139), and also Eq. (6.8.139) on β3 is separated (due to the fact that the variable β3 is cyclic). Thus, to integrate completely the system (6.8.133)–(6.8.135), it suffices to specify two independent first integrals of system (6.8.133), one by one the first integral of systems (6.8.134) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.8.135) (i.e., only n). In particular, to integrate completely the system (6.8.136)–(6.8.139), it suffices to specify two independent first integrals of system (6.8.136), one by one the first integral of systems (6.8.137), (6.8.138), and an additional first integral that “attaches” Eq. (6.8.139) (i.e., only five). If the problem on the first integrals of the system (6.8.78)–(6.8.86) (or (6.8.98)–(6.8.100)) is solved using Corollary 6.8.1, the same problem for the system (6.8.115)–(6.8.123) (or (6.8.133)–(6.8.135)) can be solved by Theorem 6.8.5 below. First, we note that one of the first integrals of the system (6.8.133) has the following form (see [199]): 2 2 (1 + bH1 )(wn−1 + wn−2 ) − (b + H1 )wn−1 sin α + sin2 α = C1 = const . wn−2 sin α (6.8.140) In the sequel, we study an additional first integral of the third-order system (6.8.133) using, in this case, the first integral (6.8.140). For this we introduce the following notation and new variables:

Θ1 (wn−1 , wn−2 ; α) =

1 . (6.8.141) τ2 Then the problem on the explicit form of the desired first integral reduces to solving the linear inhomogeneous equation wn−1 = u2 τ,

τ = sin α,

wn−2 = u1 τ,

p=

2((1 + bH1 )u2 − b)p + 2b(1 − H1 u2 − u22 − U12 (C1 , u2 )) dp = , du2 1 − (b + H1 )u2 + (1 + bH1 )u22 − (1 + bH1 )U12 (C1 , u2 )   $ 1 2 2 C1 ± C1 − 4(1 + bH1 )(1 − (b + H1 )u2 + (1 + bH1 )u2 ) ; U1 (C1 , u2 ) = 2 in this case, an additive constant C1 can be chosen as follows: (b − H1 )2 + C12 − 4 ≥ 0.

(6.8.142)

(6.8.143)

The last fact means that we can find another transcendental first integral in the explicit form (i.e., as a finite combination of quadratures). Here, the general solution of Eq. (6.8.142) depends on an arbitrary constant C2 . We omit the calculation but note that the general solution of the linear homogeneous equation obtained from (6.8.142) even in the particular case |b − H1 | = 2,

C1 =

1 − A41 , 1 + A41

A1 =

1 (b + H1 ) 2

has the following solution: (  4 4  C 2 − 4A2 (1 − A u )2 ± C ±A1 /(1+A1 ) 1 2 1 1 1 × ( 2   C1 − 4A21 (1 − A1 u2 )2 ∓ C1 

2/(1+A41 ) 

p = p0 (u2 ) = C[1 − A1 u2 ] × exp 384

2(A1 − b) , (1 + A41 )A1 (A1 u2 − 1)

C = const .

(6.8.144)

Then the desired additional first integral has the following structural form (which is similar to the transcendental first integral from the plane-parallel dynamics):  wn−1 wn−2  , = C2 = const, (6.8.145) Θ2 (wn−1 , wn−2 ; α) = G sin α, sin α sin α in this case, we use the notation and substitutions (6.8.141). Thus, we have found two first integrals (6.8.140), (6.8.145) of the independent third-order system (6.8.133). To integrate it completely, it suffices to find one by one the first integral for the systems (6.8.134) (all n − 3 pieces), and an additional first integral that “attaches” Eq. (6.8.135). Indeed, the desired first integrals coincide with the first integrals (6.8.113) and (6.8.114), namely, ( 1 + ws2  = Cs+2 = const, s = 1, . . . , n − 3, (6.8.146) Θs+2 (ws ; βs ) = sin βs Θn (wn−3 , wn−4 ; βn−4 , βn−3 , βn−2 ) = βn−2 ± arctg $

Cn−1 cos βn−3 2 2 Cn−2 sin2 βn−3 − Cn−1

= Cn = const, (6.8.147)

in this case, on the left-hand side of Eq. (6.8.147), we must substitute instead of Cn−2 , Cn−1 the first integral (6.8.146) for s = n − 4, n − 3. Theorem 6.8.5. The n first integrals (6.8.140), (6.8.145), (6.8.146), and (6.8.147) of the system (6.8.133)–(6.8.135) are transcendental functions of the phase variables and are expressed as a finite combination of elementary functions. Theorem 6.8.6. The n first integrals (6.8.140), (6.8.145), (6.8.146), and (6.8.147) of the system (6.8.133)–(6.8.135) are equivalent to n first integrals (6.8.111), (6.8.112), (6.8.113), and (6.8.114) of the system (6.8.98)–(6.8.100). Indeed, the couples of the first integrals (6.8.140), (6.8.111), (6.8.146), (6.8.113) and (6.8.147), (6.8.114) coincide if we substitute b = −b∗ and H1 = −H1∗ . We must identify the phase variables wk , k = 1, . . . , n − 1, for the system (6.8.133)–(6.8.135) with the phase variables wk , k = 1, . . . , n − 1, for the system (6.8.98)–(6.8.100). Because of their cumbersome character, similar arguments concerning the couples of the first integrals (6.8.145), (6.8.112) are not represented. Thus, we have the following topological and mechanical analogies in the sense explained above. (1) Motion of a fixed multi-dimensional physical pendulum on a (generalized) spherical hinge in a flowing medium (nonconservative force fields under assumption of additional dependence of the moment of the forces on the tensor of angular velocity). (2) Free motion of a multi-dimensional rigid body in a nonconservative force field under a tracing force (in the presence of a nonintegrable constraint under assumption of additional dependence of the moment of the forces on the tensor of angular velocity). (3) Composite motion of a multi-dimensional rigid body rotating about its center of mass, which moves rectilinearly and uniformly, in a nonconservative force field under assumption of additional dependence of the moment of the forces on the tensor of angular velocity. On more general topological analogues, see also [161, 168, 199]. REFERENCES 1. R. R. Aidagulov and M. V. Shamolin, “A certain improvement of Conway algorithm,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 3, 53–55 (2005). 2. R. R. Aidagulov and M. V. Shamolin, “Phenomenological approach to definition of interphase forces,” Dokl. Ross. Akad. Nauk, 412, No. 1, 44–47 (2007). 3. R. R. Aidagulov and M. V. Shamolin, “Archimedean uniform structure,” Sovr. Mat. Fundam. Napr., 23, 46–51 (2007). 385

4. R. R. Aidagulov and M. V. Shamolin, “General spectral approach to continuous medium dynamics,” Sovr. Mat. Fundam. Napr., 23, 52–70 (2007). 5. R. R. Aidagulov and M. V. Shamolin, “Varieties of continuous structures,” Sovr. Mat. Fundam. Napr., 23, 71–86 (2007). 6. R. R. Aidagulov and M. V. Shamolin, “Groups of colors,” in: Sovr. Mat. Prilozh., 62, 15–27 (2009). 7. R. R. Aidagulov and M. V. Shamolin, “Pseudodifferential operators in the theory of multiphase, multi-rate flows,” Sovr. Mat. Prilozh., 65, 11–30 (2009). 8. R. R. Aidagulov and M. V. Shamolin, “Averaging operators and real equations of hydromechanics,” Sovr. Mat. Prilozh., 65, 31–47 (2009). 9. R. R. Aidagulov and M. V. Shamolin, “Integration formulas of tenth order and higher,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 4, 3–7 (2010). 10. A. V. Andreev and M. V. Shamolin, “Mathematical modeling of a medium interaction onto rigid body and new two-parametric family of phase portraits,” Vestn. Samar. Univ. Estestv. Nauki, No. 10 (121), 109–115 (2014). 11. A. A. Andronov, Collection of Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1956). 12. A. A. Andronov and L. S. Pontryagin, “Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937). 13. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Oscillation Theory [in Russian], Nauka, Moscow (1981). 14. D. V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Tr. Mat. Inst. Akad. Nauk SSSR, 90 (1967). 15. P. Appel, Theoretical Mechanics [Russian translation],Vols. I, II, Fizmatgiz, Moscow (1960). 16. V. I. Arnold, “Hamiltonian property of Euler equations of rigid body dynamics in an ideal fluid,” Usp. Mat. Nauk, 24, No. 3, 225–226 (1969). 17. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989). 18. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Direction [in Russian], 3, All-Union Institute for Scientific and Technical Information (VINITI), Moscow (1985). 19. N. N. Bautin and E. A. Leontovich, Methods and Tools for Qualitative Study of Dynamical Systems on the Plane [in Russian], Nauka, Moscow (1976). 20. V. V. Beletskii, Artificial Satellite Motion with Respect to Center of Mass [in Russian], Nauka, Moscow (1965). 21. A. V. Belyaev, “On many-dimensional body motion with clumped point in gravity force field,” Mat. Sb., 114, No. 3, 465–470 (1981). 22. I. Bendixon, “On curves defined by differential equations,” Usp. Mat. Nauk, 9 (1941). 23. M. Berger, Geometry [Russian translation], Vols. I, II, Mir, Moscow (1984). 24. A. L. Besse, Manifolds with Closed Geodesics [Russian translation], Mir, Moscow (1981). 25. Yu. K. Bivin, V. V. Viktorov, and L. L. Stepanov, “Study of rigid body motion in a clayey medium,” Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 2, 159–165 (1978). 26. Yu. K. Bivin, Yu. M. Glukhov, and Yu. V. Permyakov, “Vertical entrance of rigid bodies into water,” Izv. Akad. Nauk SSSR. Mekh. Zhidkosti Gaza, No. 6, 3–9 (1985). 27. G. D. Birkhoff, Dynamical Systems [Russian translation], Gostekhizdat, Moscow–Leningrad (1941). 28. R. L. Bishop, Oscillations [Russian translation], Nauka, Moscow (1986). 29. G. A. Bliss, Lectures on the Calculus of Variations [Russian translation], Gostekhizdat, Moscow (1941).

386

30. O. I. Bogoyavlenskii, Methods of Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics [in Russian], Nauka, Moscow (1980). 31. O. I. Bogoyavlenskii, “Some integrable cases of Euler equation,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986). 32. O. I. Bogoyavlenskii and G. F. Ivakh, “Topological analysis of integrable cases of V. A. Steklov,” Usp. Mat. Nauk, 40, No. 4, 145–146 (1985). 33. G. L. Boiko and V. A. Eroshin, “Finding overloadings under a shock profile on fluid surface,” Izv. Akad. Nauk SSSR. Mekh. Zhidkosti Gaza, No. 1, 35–38 (1975). 34. I. T. Borisenok and M. V. Shamolin, “Solution of differential diagnosis problem,” Fundam. Prikl. Mat., 5, No. 3, 775–790 (1999). 35. I. T. Borisenok and M. V. Shamolin, “Solution of differential diagnosis problem by statistical trial method,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 1, 29–31 (2001). 36. A. V. Brailov, “Some cases of complete integrability of Euler equations and applications,” Dokl. Akad. Nauk SSSR, 268, No. 5, 1043–1046 (1983). 37. A. D. Bryuno, Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979). 38. N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1970). 39. N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1972). 40. A. A. Burov and G. I. Subkhankulov, “On rigid body motion in magnetic field,” Prikl. Mat. Mekh., 50, No. 6, 960–966 (1986). 41. G. S. Byushgens and R. V. Studnev, Dynamics of Longitudinal and Lateral Motion [in Russian], Mashinostroenie, Moscow (1969). 42. G. S. Byushgens and R. V. Studnev, Airplane Dynamics. Spatial Motion [in Russian], Mashinostroenie, Moscow (1988). 43. M. L. Byalyi, “On first integrals polynomial in impulses for a mechanical system on a twodimensional torus,” Funkts. Anal. Prilozh., 21, No. 4, 64–65 (1987). 44. Ch. J. de la Vall´ee Poussin, Lectures on Theoretical Mechanics [Russian translation], Vols. I, II, IL, Moscow (1948, 1949). 45. S. V. Vishik and S. F. Dolzhanskii, “Analogs of Euler–Poisson equations and magnetoelectrodynamics related to Lie groups,” Dokl. Akad. Nauk SSSR, 238, No. 5, 1032–1035. 46. F. R. Gantmakher, Lectures on Analytical Mechanics [in Russian], Nauka, Moscow (1960). 47. D. V. Georgievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with a fixed point in Rn ,” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001). 48. D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in Rn ,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002). 49. D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion for a generalized gyroscope in Rn ,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 5, 37–41 (2003). 50. D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, ‘Topical problems of geometry and mechanics’ named after V. V. Trofimov,” Sovr. Mat. Prilozh., 65, 3–10 (2009). 51. D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, ‘Topical problems of geometry and mechanics’ named after V. V. Trofimov,” Sovr. Mat. Prilozh., 76, 3–10 (2012). 52. D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies,” Sovr. Mat. Prilozh., 76, 22–39 (2012). 53. E. B. Gledzer, F. S. Dolzhanskii, and A. M. Obukhov, Systems of Hydrodynamic Type and Their Applications [in Russian], Nauka, Moscow (1981).

387

54. C. Godbillon, Differential Geometry and Analytical Mechanics [Russian translation], Mir, Moscow (1973). 55. V. V. Golubev, Lectures on Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1950). 56. V. V. Golubev, Lectures on Integrating Equations of Heavy Body Motion around a Fixed Point [in Russian], Gostekhizdat, Moscow–Leningrad (1953). 57. P. Griffits, Exterior Differential Systems and the Calculus of Variations [Russian translation], Mir, Moscow (1986). 58. D. M. Grobman, “Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962). 59. M. I. Gurevich, Jet Theory of Ideal Fluid [in Russian], Nauka, Moscow (1979). 60. B. A. Dubrovin and S. P. Novikov, “On Poisson brackets of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 279, No. 2, 294–297 (1984). 61. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Theory and Applications [in Russian], Nauka, Moscow (1979). 62. V. A. Eroshin, “Plate reflection from ideal incompressible fluid surface,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 6, 99–104 (1970). 63. V. A. Eroshin, “Submergence of a disk in a compressible fluid at an angle to a free surface,” Izv. Akad. Nauk SSSR, Mekh. Zhidkosti Gaza, No. 2, 142–144 (1983). 64. V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, “Model problem of body drag in a resisting medium under streamline flow-around,” Izv. Ross. Akad. Nauk. Mekh. Zhidkosti Gaza, No. 3, 23–27 (1995). 65. V. A. Eroshin, G. A. Konstantinov, N. I. Romanenkov, and Yu. L. Yakimov, “Experimental finding of pressure on a disk under its submergence in a compressible fluid at an angle to a free surface,” Izv. Akad. Nauk SSSR. Mekh. Zhidkosti Gaza, No. 2, 21–25 (1988). 66. V. A. Eroshin, N. I. Romanenkov, I. V. Serebryakov, and Yu. L. Yakimov, “Hydrodynamic forces under a shock of blunt bodies on compressible fluid surface,” Izv. Akad. Nauk SSSR. Mekh. Zhidkosti Gaza, No. 6, 44–51 (1980). 67. N. E. Zhukovskii, “On a fall of light oblong bodies rotating around their longitudinal axis,” in: Complete Collection of Works [in Russian], Vol. 5, Fizmatgiz, Moscow (1937), pp. 72–80, 100–115. 68. N. E. Zhukovskii, “On bird soaring,” in: Complete Collection of Works [in Russian], Vol. 5, Fizmatgiz, Moscow (1937), pp. 49–59. 69. H. Seifert and W. Threifall, Topology [Russian translation], Gostekhizdat, Moscow–Leningrad (1938). 70. T. A. Ivanova, “On Euler equations in models of theoretical physics,” Mat. Zametki, 52, No. 2, 43–51 (1992). 71. A. Yu. Ishlinsky, Orientation, Gyroscopes, and Inertial Navigation [in Russian], Nauka, Moscow (1976). 72. V. V. Kozlov, Qualitative Analysis Methods in Rigid Body Dynamics [in Russian], Izd. Mosk. Univ., Moscow (1980). 73. V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983). 74. V. V. Kozlov, “On rigid body fall in ideal fluid,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 10–17 (1989). 75. V. V. Kozlov, “To problem of heavy rigid body fall in a resisting medium,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 1, 79–87 (1990). 76. G. Lamb, Hydrodynamics [Russian translation], Fizmatgiz, Moscow (1947).

388

77. S. Lefschetz, Geometric Theory of Differential Equations [Russian translation], IL, Moscow (1961). 78. J. U. Leech, Classical Mechanics [Russian translation], IL, Moscow (1961). 79. B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, An Introduction to Problem of Motion of a Body in a Resisting Medium [in Russian], Izd. Mosk. Univ., Moscow (1986). 80. V. V. Lunev, “A hydrodynamical analogy of problem of rigid body motion with a fixed point in Lorenz force field,” Dokl. Akad. Nauk SSSR, 276, No. 2, 351–355 (1984). 81. A. M. Lyapunov, “A new integrability case of equations of rigid body motion in a fluid,” in: Collection of Works [in Russian], Vol. I, Izd. Akad. Nauk SSSR, Moscow (1954), pp. 320–324. 82. Yu. I. Manin, “Algebraic aspects of theory of nonlinear differential equations,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian], 11, AllUnion Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1978), pp. 5–112. 83. W. Miller, Symmetry and Separation of Variables [Russian translation], Mir, Moscow (1981). 84. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1949). 85. Z. Nitetski, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975). 86. V. A. Odareev, Decompositional Analysis of Dynamics and Stability of Longitudinal Perturbed Motion of the Ekranoplan [in Russian], Doctoral Dissertation, MGAI, Moscow (1995). 87. Yu. M. Okunev and V. A. Sadovnichii, “Model dynamical systems of a certain problem of external ballistics and their analytical solutions,” in: Problems of Modern Mechanics [in Russian], Izd. Mosk. Univ., Moscow (1998), pp. 28–46. 88. Yu. M. Okunev, V. A. Samsonov, B. Ya. Lokshin, M. Z. Dosaev, L. A. Klimina, Yu. D. Selyutskii, O. G. Privalova, M. V. Shamolin, and A. I. Kobrin, “Problems of control of motion of the bodies in a continuous medium,” in: Scientific Report of Institute of Mechanics, Moscow State University [in Russian], No. 5103, Institute of Mechanics, Moscow State University, Moscow (2010). 89. Yu. M. Okunev and M. V. Shamolin, “On integrability in elementary functions of certain classes of complex nonautonomous equations,” Sovr. Mat. Prilozh., 65, 122–131 (2009). 90. J. Palis and W. De Melu, Geometric Theory of Dynamical Systems. An Introduction [Russian translation], Mir, Moscow (1986). 91. A. M. Perelomov, “Several remarks on integration of equations of rigid body motion in ideal fluid,” Funkts. Anal. Prilozh., 15, No. 2, 83–85 (1981). 92. N. L. Polyakov and M. V. Shamolin, “On closed symmetric functional classes preserving any single predicate,” Vestn. Sam. Univ. Estestv. Nauki, No. 6 (107), 61–73 (2013). 93. N. L. Polyakov and M. V. Shamolin, “On a generalization of Arrows impossibility theorem,” Dokl. Ross. Akad. Nauk, 456, No. 2, 143–145 (2014). 94. N. V. Pokhodnya and M. V. Shamolin, “New integrable case in dynamics of a multidimensional body,” Vestn. Sam. Univ. Estestv. Nauki, No. 9 (100), 136–150 (2012). 95. N. V. Pokhodnya and M. V. Shamolin, “Certain conditions of integrability of dynamical systems in transcendental functions,” Vestn. Sam. Univ. Estestv. Nauki, No. 9/1 (110), 35–41 (2013). 96. N. V. Pokhodnya and M. V. Shamolin, “Integrable systems on the tangent bundle of a multidimensional sphere,” Vestn. Sam. Univ. Estestv. Nauki, No. 7 (118), 60–69 (2014). 97. H. Poincar´e, On Curves Defined by Differential Equations [Russian translation], OGIZ, Moscow– Leningrad (1947). 98. H. Poincar´e, “New methods in celestial mechanics,” in: Selected Works [Russian translation], Vols. 1, 2, Nauka, Moscow (1971, 1972). 99. H. Poincar´e, On Science [Russian translation], Nauka, Moscow (1983).

389

100. V. A. Samsonov, Issues on Mechanics. Some Problems, Phenomena, and Paradoxes [in Russian], Nauka, Moscow (1980). 101. V. A. Samsonov and M. V. Shamolin, “Problem on body motion in a resisting medium,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 3, 51–54 (1989). 102. V. A. Samsonov and M. V. Shamolin, “A model problem of body motion in a medium with streamline flow-around,” Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 3969, Institute of Mechanics, Moscow State Univ., Moscow (1990). 103. V. A. Samsonov and M. V. Shamolin, “Problem of body drag in a medium under streamline flow-around,” in: Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4141, Institute of Mechanics, Moscow State Univ., Moscow (1991). 104. V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin, “Mathematical modelling in problem of body drag in a resisting medium under streamline flow-around,” in: Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4396, Institute of Mechanics, Moscow State Univ., Moscow (1995). 105. L. I. Sedov, Mechanics of Continuous Media [in Russian], Vols. 1, 2, Nauka, Moscow (1983, 1984). 106. N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a certain problem with free boundary,” Vestn. Sam. Univ. Estestv. Nauki, No. 8 (89), 86–94 (2011). 107. N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a one-phase problem with free boundary,” Sovr. Mat. Prilozh., 78, 99–108 (2012). 108. N. Yu. Selivanova and M. V. Shamolin, “Studying the interphase zone in a certain singular-limit problem,” Sovr. Mat. Prilozh., 78, 109–118 (2012). 109. N. Yu. Selivanova and M. V. Shamolin, “Local solvability of the capillar problem,” Sovr. Mat. Prilozh., 78, 119–125 (2012). 110. N. Yu. Selivanova and M. V. Shamolin, “Quasi-stationary Stefan problem with values on front depending on its geometry,” Sovr. Mat. Prilozh., 78, 126–134 (2012). 111. J. L. Singh, Classical Dynamics [Russian translation], Fizmatgiz, Moscow (1963). 112. S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970). 113. V. A. Steklov, On Rigid Body Motion in a Fluid [in Russian], Kharkov (1893). 114. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946). 115. V. V. Sychev, A. I. Ruban, Vik. V. Sychev, and G. L. Korolev, Asymptotic Theory of Separation Flows [in Russian], Nauka, Moscow (1987). 116. V. G. Tabachnikov, “Stationary characteristics of wings in small velocities under whole range of attack angles,” in: Proc. Central Aero-Hydrodynamical Institute [in Russian], No. 1621, Moscow (1974), pp. 18–24. 117. V. V. Trofimov, “Embeddings of finite groups in compact Lie groups by regular elements,” Dokl. Akad. Nauk SSSR, 226, No. 4, 785–786 (1976). 118. V. V. Trofimov, “Euler equations on finite-dimensional solvable Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 5, 1191–1199 (1980). 119. V. V. Trofimov and M. V. Shamolin, “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems,” Fundam. Prikl. Mat., 16, No. 4, 3–229 (2010). 120. E. T. Whittecker, Analytical Dynamics [Russian translation], ONTI, Moscow (1937). 121. P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970). 122. S. A. Chaplygin, “On motion of heavy bodies in an incompressible fluid,” in: Complete Collection of Works [in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133–135. 123. S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976). 124. M. V. Shamolin, Qualitative Analysis of a Model Problem of Body Motion in a Medium with Streamline Flow-Around [in Russian], Candidate Dissertation, Moscow Univ., Moscow (1991).

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125. M. V. Shamolin, “Closed trajectories of different topological types in problem of body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 2, 52–56 (1992). 126. M. V. Shamolin, “Problem of body motion in a medium with resistance,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 1, 52–58 (1992). 127. M. V. Shamolin, “Classification of phase portraits in problem of body motion in a resisting medium in the presence of a linear damping moment,” Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993). 128. M. V. Shamolin, “Applications of Poincar´e topographical system methods and comparison systems in some concrete systems of differential equations,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 2, 66–70 (1993). 129. M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on a plane,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 1, 68–71 (1993). 130. M. V. Shamolin, “A new two-parameter family of phase portraits in problem of a body motion in a medium,” Dokl. Ross. Akad. Nauk, 337, No. 5, 611–614 (1994). 131. M. V. Shamolin, “Relative structural stability of dynamical systems for problem of body motion in a medium,” in: Analytical, Numerical, and Experimental Methods in Mechanics. A Collection of Scientific Works [in Russian], Izd. Mosk. Univ., Moscow (1995), pp. 14–19. 132. M. V. Shamolin, “Definition of relative roughness and two-parameter family of phase portraits in rigid body dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996). 133. M. V. Shamolin, “Periodic and Poisson stable trajectories in problem of body motion in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55–63 (1996). 134. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996). 135. M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new twoparameter family of phase portraits,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 4, 57–69 (1996). 136. M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997). 137. M. V. Shamolin, “Spatial Poincar´e topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). 138. M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). 139. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: CD Proc. of the Congress “Nonlinear Analysis and Its Applications,” Moscow, Russia, September 1–5, 1998, Moscow (1999), pp. 497–508. 140. M. V. Shamolin, “Families of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998). 141. M. V. Shamolin, “Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178–189 (1999). 142. M. V. Shamolin, “New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999). 143. M. V. Shamolin, “On roughness of dissipative systems and relative roughness and nonroughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999). 144. M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000). 145. M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000).

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146. M. V. Shamolin, “Jacobi integrability in problem of four-dimensional rigid body motion in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000). 147. M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 189–193 (2001). 148. M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,” Prikl. Mekh., 37, No. 6, 74–82 (2001). 149. M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in a jet flow of medium,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 5, 22–28 (2001). 150. M. V. Shamolin, “On integrability of certain classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). 151. M. V. Shamolin, “Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004). 152. M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Doctoral Dissertation, Moscow Univ., Moscow (2004). 153. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2004). 154. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking account of rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005). 155. M. V. Shamolin, “Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow-around,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005). 156. M. V. Shamolin, “On a certain integrable case of equations of dynamics in so(4) × R4 ,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). 157. M. V. Shamolin, “Model problem of body motion in a resisting medium taking account of dependence of resistance force on angular velocity,” in: Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4818, Institute of Mechanics, Moscow State Univ., Moscow (2006). 158. M. V. Shamolin, “Problem on rigid body spatial drag in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 45–57 (2006). 159. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in medium flow taking account of rotational derivatives of moments of its action force,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 187–192 (2007). 160. M. V. Shamolin, “A case of complete integrability in dynamics on a tangent bundle of twodimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007). 161. M. V. Shamolin, Methods of Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). 162. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2007). 163. M. V. Shamolin, “Three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008). 164. M. V. Shamolin, “Some model problems of dynamics for a rigid body interacting with a medium,” Prikl. Mekh., 43, No. 10, 49–67 (2007). 165. M. V. Shamolin, “New integrable cases in dynamics of a medium-interacting body with allowance for dependence of resistance force moment on angular velocity,” Prikl. Mat. Mekh., 72, No. 2, 273–287 (2008). 166. M. V. Shamolin, “Integrability of some classes of dynamic systems in terms of elementary functions,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 3, 43–49 (2008).

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167. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Sovr. Mat. Prilozh., 62, 131–171 (2009). 168. M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications,” Fundam. Prikl. Mat., 14, No. 3, 3–237 (2008). 169. M. V. Shamolin, “New cases of full integrability in dynamics of a dynamically symmetric fourdimensional solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 425, No. 3, 338–342 (2009). 170. M. V. Shamolin, “Generalized problem of differential diagnosis and its possible resolving,” Elektron. Model., 31, No. 1, 97–115 (2009). 171. M. V. Shamolin, “Resolving of diagnosis problem in case of precise trajectory measurements with error,” Elektron. Model., 31, No. 3, 73–90 (2009). 172. M. V. Shamolin, “Diagnosis of failures in certain non-direct control system,” Elektron. Model., 31, No. 4, 55–66 (2009). 173. M. V. Shamolin, “Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field,” Sovr. Mat. Prilozh., 65, 132–142 (2009). 174. M. V. Shamolin, “Stability of a rigid body translating in a resisting medium,” Prikl. Mekh., 45, No. 6, 125–140 (2009). 175. M. V. Shamolin, “New cases of integrability in the spatial dynamics of a rigid body,” Dokl. Ross. Akad. Nauk, 431, No. 3, 339–343 (2010). 176. M. V. Shamolin, “A completely integrable case in the dynamics of a four-dimensional rigid body in a non-conservative field,” Usp. Mat. Nauk, 65, No. 1, 189–190 (2010). 177. M. V. Shamolin, “Diagnosis of certain system of direct control of aircraft motion,” Elektron. Model., 32, No. 1, 45–52 (2010). 178. M. V. Shamolin, “On the problem of the motion of the body with plane front end in a resisting medium,” in: Scientific Report of Institute of Mechanics, Moscow State University [in Russian], No. 5052, Institute of Mechanics, Moscow State University, Moscow (2010). 179. M. V. Shamolin, “Spatial motion of a rigid body in a resisting medium,” Prikl. Mekh., 46, No. 7, 120–133 (2010). 180. M. V. Shamolin, “Motion diagnosis of aircraft in mode of planning lowering,” Elektron. Model., 32, No. 5, 31–44 (2010). 181. M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,” Sovr. Mat. Prilozh., 76, 84–99 (2012). 182. M. V. Shamolin, “A new case of integrability in dynamics of a 4D-solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 437, No. 2, 190–193 (2011). 183. M. V. Shamolin, “Complete list of first integrals in the problem on the motion of a 4D solid in a resisting medium under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 440, No. 2, 187–190 (2011). 184. M. V. Shamolin, “Diagnosis of hyro-stabilized platform included in control system of aircraft motion,” Elektron. Model., 33, No. 3, 121–126 (2011). 185. M. V. Shamolin, “Dynamical invariants of integrable variable dissipation dynamical systems,” Vestn. Nizhegorod. Univ., Part 2, No. 4, 356–357 (2011). 186. M. V. Shamolin, “A multiparameter family of phase portraits in the dynamics of a rigid body interacting with a medium,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 3, 24–30 (2011). 187. M. V. Shamolin, “Rigid body motion in a resisting medium,” Mat. Model., 23, No. 12, 79–104 (2011). 188. M. V. Shamolin, “A new case of integrability in spatial dynamics of a rigid solid interacting with a medium under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 442, No. 4, 479–481 (2012).

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189. M. V. Shamolin, “New case of complete integrability of the dynamic equations on the tangent stratification of three-dimensional sphere,” Vestn. Sam. Univ. Estestv. Nauki, No. 5 (86), 187– 189, (2011). 190. M. V. Shamolin, “A new case of integrability in the dynamics of a 4D-rigid body in a nonconservative field under the assumption of linear damping,” Dokl. Ross. Akad. Nauk, 444, No. 5, 506–509 (2012). 191. M. V. Shamolin, “Some questions of qualitative theory in dynamics of systems with variable dissipation,” in: Sovr. Mat. Prilozh., 78, 138–147 (2012). 192. M. V. Shamolin, “Complete list of first integrals of dynamic equations of spatial rigid body motion in a resisting medium under assumption of linear damping,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 4, 44–47 (2012). 193. M. V. Shamolin, “The problem of a rigid body motion in a resisting medium with the assumption of dependence of the force moment on the angular velocity,” Mat. Model., 24, No. 10, 109–132 (2012). 194. M. V. Shamolin, “Complete list of first integrals of dynamic equations of motion of a 4D rigid body in a nonconservative field under the assumption of linear damping,” Dokl. Ross. Akad. Nauk, 449, No. 4, 416–419 (2013). 195. M. V. Shamolin, “A new case of integrability in transcendental functions in the dynamics of solid body interacting with the environment,” Avtomat. Telemekh., 8, 173–190 (2013). 196. M. V. Shamolin, “New case of integrability in the dynamics of a multidimensional solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 453, No. 1, 46–49 (2013). 197. M. V. Shamolin, “New case of integrability of dynamic equations on the tangent bundle of a 3-sphere,” Usp. Mat. Nauk, 68, No. 5, 185–186 (2013). 198. M. V. Shamolin, “On integrability in dynamic problems for a rigid body interacting with a medium,” Prikl. Mekh., 49, No. 6, 44–54 (2013). 199. M. V. Shamolin, “Variety of integrable cases in dynamics of low- and multi-dimensional rigid bodies in nonconservative force fields,” in: Itogi Nauki i Tekh. Ser. Sovr. Mat. Prilozh. Temat. Obzory [in Russian], 125, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2013), pp. 5–254. 200. M. V. Shamolin, “A new case of integrability in the dynamics of a multidimensional solid in a nonconservative field under the assumption of linear damping,” Dokl. Ross. Akad. Nauk, 457, No. 5, 542–545 (2014). 201. M. V. Shamolin, “Variety of integrable cases in spatial dynamics of a rigid body in a nonconservative force fields,” in: Tr. Sem. I. G. Petrovskogo, 30, Moscow State Univ., Moscow (2014), pp. 287–350. 202. M. V. Shamolin, “A multidimensional pendulum in a nonconservative force field,” Dokl. Ross. Akad. Nauk, 460, No. 2, 165–169 (2015). 203. M. V. Shamolin, “Rigid body motion in a resisting medium modelling and analogues with vortex streets,” Mat. Model., 27, No. 1, 33–53 (2015). 204. M. V. Shamolin, “Integrable cases in the dynamics of a multi-dimensional rigid body in a nonconservative field in the presence of a tracking force,” Fundam. Prikl. Mat., 19, No. 3, 187–222 (2014). 205. M. V. Shamolin, “Complete list of first integrals of dynamic equations for a multidimensional solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 461, No. 5, 533–536 (2015). 206. M. V. Shamolin, “New case of complete integrability of dynamics equations on a tangent bundle to the three-dimensional sphere”, Vestn. Mosk. Univ. Ser. 1, Mat. Mekh., 3, 11–14 (2015). 207. M. V. Shamolin and S. V. Tsyptsyn, “Analytical and numerical study of trajectories of body motion in a resisting medium,” in: Scientific Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4289, Institute of Mechanics, Moscow State Univ., Moscow (1993). 394

208. M. V. Shamolin and D. V. Shebarshov, “Some problems of geometry in classical mechanics,” Preprint VINITI No. 1499–V99 (1999). 209. M. V. Shamolin and D. V. Shebarshov, “Methods for solving main problem of differential diagnosis,” Preprint VINITI, No. 1500–V99 (1999). 210. O. P. Shorygin and N. A. Shulman, “Entry of a disk into water at an attack angle,” Uch. Zap. TsAGI, 8, No. 1, 12–21 (1977). 211. D. Arrowsmith and C. Place, Ordinary Differential Equations. Qualitative Theory with Applications [Russian translation], Mir, Moscow (1986). 212. C. Jacobi, Lectures on Dynamics [Russian translation], ONTI, Moscow (1936). 213. R. R. Aidagulov and M. V. Shamolin, “Polynumbers, norms, metrics, and polyingles,” J. Math. Sci., 204, No. 6, 742–759 (2015). 214. R. R. Aidagulov and M. V. Shamolin, “Finsler spaces, bingles, polyingles, and their symmetry groups,” J. Math. Sci., 204, No. 6, 732–741 (2015). 215. R. R. Aidagulov and M. V. Shamolin, “Topology on polynumbers and fractals,” J. Math. Sci., 204, No. 6, 760–771 (2015). 216. D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University ‘Urgent problems of geometry and mechanics’ named after V. V. Trofimov,” J. Math. Sci., 154, No. 4, 462–495 (2008). 217. D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University ‘Urgent problems of geometry and mechanics’ named after V. V. Trofimov,” J. Math. Sci., 161, No. 5, 603–614 (2009). 218. D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University ‘Urgent problems of geometry and mechanics’ named after V. V. Trofimov,” J. Math. Sci., 204, No. 6, 715–731 (2015). 219. Yu. M. Okunev and M. V. Shamolin, “On the Construction of the General Solution of a Class of Complex Nonautonomous Equations,” J. Math. Sci., 204, No. 6, 787–799 (2015). 220. M. V. Shamolin, “Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in: Proc. WCSMO-2, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 387–392. 221. M. V. Shamolin, “Some classical problems in three-dimensional dynamics of a rigid body interacting with a medium,” in: Proc. ICTACEM98, Kharagpur, India, December 1–5, 1998, Indian Inst. Technology, Kharagpur, India (1998), pp. 1–11. 222. M. V. Shamolin, “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” in: CD Proc. ECCOMAS 2000, Barcelona, Spain, September 11–14, Barcelona (2000), pp. 1–11. 223. M. V. Shamolin, “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002). 224. M. V. Shamolin, “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium,” J. Math. Sci., 114, No. 1, 919–975 (2003). 225. M. V. Shamolin, “Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003). 226. M. V. Shamolin, “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004). 227. M. V. Shamolin, “Structural stable vector fields in rigid body dynamics,” in: Proc. 8th Conf. Dynamical Systems: Theory and Applications (DSTA 2005), Lodz, Poland, December 12–15, 2005, 1, Technical Univ. Lodz (2005), pp. 429–436. 228. M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Proc. 9th Conf. Dynamical Systems: Theory 395

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and Applications (DSTA 2007), Lodz, Poland, December 17–20, 2007, 1, Technical Univ. Lodz (2007), pp. 415–422. M. V. Shamolin, “Some methods of analysis of the dynamic systems with various dissipation in dynamics of a rigid body,” Proc. Appl. Math. Mech., 8, 10137–10138 (2008). M. V. Shamolin, “Dynamical systems with variable dissipation: Methods and applications,” in: Proc. 10th Conf. Dynamical Systems: Theory and Applications (DSTA 2009), Lodz, Poland, December 7–10, 2009, Technical Univ. Lodz (2009), pp. 91–104. M. V. Shamolin, “The various cases of complete integrability in dynamics of a rigid body interacting with a medium,” in: CD Proc. Conf. Multibody Dynamics, ECCOMAS Thematic Conf. Warsaw, Poland, June 29–July 2, 2009, Polish Acad. Sci., Warsaw (2009), pp. 1–20. M. V. Shamolin, “New cases of integrability in dynamics of a rigid body with the cone form of its shape interacting with a medium,” Proc. Appl. Math. Mech., 9, No. 1, 139–140 (2009). M. V. Shamolin, “Dynamical systems with various dissipation: Background, methods, applications,” in: CD Proc. XXXVIII Summer School-Conf. “Advanced Problems in Mechanics” (APM 2010), July 1–5, 2010, St. Petersburg (Repino), Russia [in Russian], St. Petersburg (2010), pp. 612–621. M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid body,” Proc. Appl. Math. Mech., 10, No. 1, 63–64 (2010). M. V. Shamolin, “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices,” in: CD Proc. 5th Int. Sci. Conf. Physics and Control PHYSCON 2011, Leon, Spain, September 5–8, 2011, Leon, Spain (2011) pp. 1–5. M. V. Shamolin, “Variety of the cases of integrability in dynamics of a 2D-, 3D-, and 4Drigid body interacting with a medium,” in: Proc. 11th Conf. Dynamical Systems: Theory and Applications (DSTA 2011), Lodz, Poland, December 5–8, 2011, Technical Univ. Lodz (2011), pp. 11–24. M. V. Shamolin, “Cases of Complete Integrability in Transcendental Functions in Dynamics and Certain Invariant Indices,” Proc. Appl. Math. Mech., 12, No. 1, 43–44 (2012). M. V. Shamolin, “Cases of integrability in transcendental functions in 3D Dynamics of a rigid body interacting with a medium,” in: Proc. ECCOMAS Multibody Dynamics 2013, Zagreb, Croatia, July 1–4, 2013, Zagreb, University of Zagreb (2013), pp. 903–912. M. V. Shamolin, “Variety of the cases of integrability in Dynamics of a symmetric 2D-, 3D- and 4D-rigid body in a nonconservative field,” Int. J. Struct. Stabil. Dynam., 13, No. 7, 1340011– 1340024 (2013). M. V. Shamolin, “Review of cases of integrability in dynamics of lower- and multidimensional rigid body in a nonconservative field of forces,” in: Proc. 2014 Int. Conf. on Pure Mathematics and Applied Mathematics (PMAM’14), Venice, Italy, March 15–17, 2014, Venice (2014), pp. 86– 102. M. V. Shamolin, “New cases of integrability in multidimensional dynamics in a nonconservative field,” in: CD Proc. XLII Summer School-Conf. “Advanced Problems in Mechanics” (APM 2014), June 30–July 5, 2014, St. Petersburg (Repino), Russia [in Russian], St. Petersburg (2014), pp. 435–446. M. V. Shamolin, “Dynamical pendulum-like nonconservative systems,” in: Applied Non-Linear Dynamical Systems, Springer Proc. Math. Stat., 93, 503–525 (2014). M. V. Shamolin, “On stability of certain key types of rigid body motion in a nonconservative field,” Proc. Appl. Math. Mech., 14, No. 1, 311–312 (2014). M. V. Shamolin, “Classification of integrable cases in the dynamics of a four-dimensional rigid body in a nonconservative field in the presence of a tracking force,” J. Math. Sci., 204, No. 6, 808–870 (2015).

245. M. V. Shamolin, “Certain Integrable Cases in Dynamics of a Multi-Dimensional Rigid Body in a Nonconservative Field,” in: Proc. Int. Conf. on Pure and Appl. Math. (PMAM’15), Vienna, Austria, March 15–17, 2015, Vienna (2015), pp. 328–342. 246. M. V. Shamolin, “Multidimensional pendulum in a nonconservative force field,” in: CD Proc. XLIII Summer School-Conf. “Advances Problems in Mechanics” (APM 2015), June 22–27, 2015, St. Petersburg, Russia [in Russian], St. Petersburg (2015), pp. 322–332. M. V. Shamolin Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: [email protected]

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