INTEGRABLE SYSTEMS ON THE TANGENT

DOI 10.1007/s10958-018-4028-1 Journal of Mathematical Sciences, Vol. 234, No. 4, October, 2018

INTEGRABLE SYSTEMS ON THE TANGENT BUNDLE OF A MULTI-DIMENSIONAL SPHERE M. V. Shamolin

UDC 517.9+531.01

Abstract. This paper contains a systematic exposition of some results on the equations of motion of a dynamically symmetric n-dimensional rigid body in a nonconservative field of forces. Similar bodies are considered in the dynamics of actual rigid bodies interacting with a resisting medium under the conditions of jet flow past the body with a nonconservative following force acting on the body in such a way that its characteristic point has a constant velocity, which means that the system has a nonintegrable servo-constraint.

Introduction This work continues the studies on the integration of the dynamical part of the equations of motion of a three-dimensional (3D-) rigid body in a field of forces that has been constructed under the assumption of quasi-stationary interaction between the body and the medium [19, 25, 27–29]. It would be relevant to mention some other publications of the author on the integration of similar equations for two-dimensional (2D-) [32, 33] and four-dimensional (4D-) [37, 42, 46, 50] rigid bodies in a nonconservative field of forces. It is hardly possible to construct any kind of integration theory for general nonconservative systems (even lower-dimensional ones). However, in some cases, for systems with additional symmetries, one can find first integrals in terms of finite combinations of elementary functions. We have found new cases of integrability of nonconservative dynamical systems with nontrivial symmetries. In almost all these cases, each first integral is expressed through a finite combination of elementary functions and, at the same time, is a transcendental function of its variables. The term transcendental is understood here in the sense of complex analysis: being extended to the complex region, the function has essential singularities. This fact is due to the presence of attractive and repelling limit sets in the system (for instance, attractive and repelling foci or nodes, limit cycles) (see also [1, 15, 16, 18, 20, 21]). Previously [44,45], we studied systems on the tangent bundle of the two-dimensional sphere and found a fairly general form of such systems of the third order with transcendental first integrals [35, 36, 43]. Many problems of multi-dimensional dynamics involve mechanical systems whose configuration space coincides with a finite-dimensional sphere. Accordingly, phase spaces of such systems are tangent bundles of those spheres. Thus, a physical pendulum on a cylindrical hinge in a plane-parallel field of forces can be considered on its phase cylinder, while studying a spatial (3D) pendulum on a spherical hinge leads to a dynamical system on the tangent bundle of a 2D sphere. The author previously considered some dynamical problems for an n-dimensional rigid body in a nonconservative force field. These problems lead to systems on the tangent bundle of an (n − 1)-dimensional sphere. In the present paper, we pay close attention to the inductive transition from systems on tangent bundles of smaller-dimensional spheres to systems on tangent bundles of spheres of an arbitrary dimension. Our investigation starts with systems without a force field and proceeds to systems with certain nonconservative force fields (see also [24, 30, 31, 48, 49]). Previously [19, 45], the author established complete integrability of the equations of plane-parallel motion of a body in a resisting medium under the conditions of jet stream past the body, in which case the system of dynamic equations admits a first integral represented by a transcendental (i.e., having Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 31, pp. 257–323, 2016.

548

c 2018 Springer Science+Business Media, LLC 1072–3374/18/2344–0548

essential singularities as a function of complex variable) function of quasi-velocities. It was assumed that the body can interact with the medium only on a portion of its surface in the shape of a (one-dimensional) plate. Later [40, 47, 53, 58, 60–62], the plane problem was extended to the spatial (3D) case, with the system of dynamic equations admitting a complete set of transcendental first integrals. Here, it was assumed that the interaction between the body and the medium is concentrated on the part of its surface in the shape of a plane (2D) disk. Further investigations [51, 52, 54–57, 59] pertain to the dynamical part of the equations of motion of various dynamically symmetric four-dimensional rigid bodies, with the field of forces concentrated on the part of the body’s surface in the shape of a two-dimensional (three-dimensional) disk, while the action of the forces is concentrated on a two-dimensional plane (one-dimensional line) orthogonal to the said disk. The results of the present paper pertain to the case in which all interaction of the body and the medium is concentrated on a part of its surface having the shape of an (n − 1)-dimensional disk, while the action of the forces is orthogonal to that disk. These results are presented in a systematic way and stated in invariant form. 1. Some General Considerations 1.1. Cases of Dynamical Symmetry of Multi-Dimensional Bodies. Consider an n-dimensional rigid body Θ of mass m with a smooth (n − 1)-dimensional boundary ∂Θ. The body is placed in a field of forces (nonconservative, in general). This can be interpreted as motion of a body in a resisting medium occupying an n-dimensional domain in the Euclidean space En ). Suppose that the body is dynamically symmetric. For instance, for a four-dimensional body there are two logically possible ways to represent its inertia tensor in the case of two independent identities of its principal moments of inertia: in some coordinate system Dx1 x2 x3 x4 fixed to the body, either the inertia operator has the form (1) diag{I1 , I2 , I2 , I2 } (the so-called case (1–3)), or it has the form diag{I1 , I1 , I3 , I3 }

(2)

(case (2–2)). In the first case, the body is dynamically symmetric in the hyperplane Dx2 x3 x4 (in other words, Dx1 is an axis of symmetry). In the second case, the two-dimensional planes Dx1 x2 and Dx3 x4 are planes of dynamic symmetry of the body. For a five-dimensional body, it would be logical to consider the cases of three independent identities of its principal moments of inertia: in some coordinate frame Dx1 x2 x3 x4 x5 fixed to the body, either the inertia operator has the form (3) diag{I1 , I2 , I2 , I2 , I2 } (case (1–4)), or it has the form diag{I1 , I1 , I3 , I3 , I3 }

(4)

(case (2–3)). In the first case, the body is dynamically symmetric in the hyperplane Dx2 x3 x4 x5 (in other words, Dx1 is an axis of dynamical symmetry). In the second case, the two-dimensional and the three-dimensional planes Dx1 x2 and Dx3 x4 x5 are planes of dynamical symmetry of the body. Accordingly, for an n-dimensional body, it would be logical to consider the cases of n − 1 independent identities of its principal moments of inertia. There are [n/2] (here [. . .] denotes the integer part) alternatives of the form (1), (2) (or (3), (4)). Thus, for a 6-dimensional body, three cases are possible: (1–5), (2–4), (3–3). With regard to an n-dimensional rigid body, we are mainly interested in the case (1–(n − 1)), i.e., in a coordinate frame Dx1 . . . xn fixed to the body, the inertia operator has the form diag{I1 , I2 , . . . , I2 },

(5) 549

and it is in the hyperplane Dx2 . . . xn that the body is dynamically symmetric (in other words, Dx1 is an axis of dynamical symmetry). 1.2. Dynamics on so(n) and Rn. The configuration space of a free n-dimensional rigid body is the direct product of Rn (corresponding to its center of mass) and the group of its rotations SO(n) (corresponding to its rotations about its center of mass), Rn × SO(n).

(6)

The dimension of this space is equal to n(n + 1) n(n − 1) = . 2 2 Accordingly, the dimension of the phase space is equal to n(n + 1). In particular, if Ω is the angular velocity tensor of an n-dimensional rigid body (a tensor of rank two [7–9]), Ω ∈ so(n), then the part of the dynamic equations of motion corresponding to the Lie algebra so(n) has the form [3, 4, 11, 13, 22, 23] ˙ + ΛΩ ˙ + [Ω, ΩΛ + ΛΩ] = M, ΩΛ (7) n+

where (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 moment of external forces F acting on the body in Rn and projected to the natural coordinates in the Lie algebra so(n), and [ , ] is the commutator in so(n). Thus, we represent the skew-symmetric matrix (associated with the said tensor of rank 2) Ω ∈ so(5) in the form ⎞ ⎛ 0 −ω10 ω9 −ω7 ω4 ⎜ ω10 0 −ω8 ω6 −ω3 ⎟ ⎟ ⎜ ⎟, ⎜−ω9 ω 0 −ω ω (9) 8 5 2 ⎟ ⎜ ⎝ ω7 −ω6 ω5 0 −ω1 ⎠ −ω4 ω3 −ω2 ω1 0 where ω1 , ω2 , . . . , ω10 are the components of the angular velocity tensor with respect to the coordinates in the Lie algebra so(5). It is obvious that here we have λi − λj = Ij − Ii

(10)

for i, j = 1, . . . , n. When calculating the moment of external forces acting on the body, it is necessary to connstruct the mapping (11) Rn × Rn → so(n) that maps a pair of vectors (12) (DN, F) ∈ Rn × Rn n n from R × R to an element of the Lie algebra so(n), where DN = {δ1 , δ2 , . . . , δn },

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

(13)

F is the external force acting on the body, and DN is the vector issuing from the origin D of the coordinate system Dx1 . . . xn and ending at the point N at which the force is applied. Here, one constructs the corresponding auxiliary matrix δ 1 δ 2 . . . δn . (14) F1 F2 . . . Fn 550

All minors of order 2 (their number is equal to n(n−1)/2), taken with the sign of this matrix, coincide with the coordinates of the moment (DN, F) of the force F, while the moment itself is identified with some element of the Lie algebra so(n). Since the coordinates ω1 , ω2 , . . . , ωn in the Lie algebra so(n) are ordered, we introduce the same ordering for calculating the moment (DN, F) of the force F. Indeed, the first group, G1 , of the coordinates of the said 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 group of the coordinates, G2 , consists of n − 2 alternating minors

δ1 δn−1

δn−2 δn−1

δn−3 δn−1

δn−4 δn−1

n+1

, −

+

F1 Fn−1 .

Fn−3 Fn−1 , + Fn−4 Fn−1 , . . . , (−1) Fn−2 Fn−1

Further, assume that the last group of the coordinates, Gn−1 , consists of a single minor

δ1 δ2

. +

F1 F2

We see that the first minors in each group have the sign “+.” The resulting ordered set of n(n − 1)/2 quantities is called the coordinates of the moment (DN, F) of the force F. The dynamical systems studied below are nonconservative, in general, and belong to the class of dynamical systems with variable zero-mean dissipation [39,45,46]. In this connection, we have to examine “directly” a part of the principal equation of motion, in our case, Newton’s equation, which takes the form of the equation of motion of the center of mass, i.e., the part of the dynamic equations of motion corresponding to Rn : mwC = F, (15) where wC is the acceleration of the center of mass C of the body, and m is its mass. Moreover, the multi-dimensional Rivals’ formula (in this case, it can be easily derived by the operator method) ensures that we also have the relations wC = wD + Ω2 DC + EDC,

wD = v˙ D + ΩvD ,

˙ E = Ω,

(16)

where wD is the acceleration of the point D; F is the external force acting on the body (in our case, F = S); E is the angular acceleration tensor (of rank 2). Suppose that the position of the body Θ in the Euclidean space En is determined by functions which are cyclic in the following sense: the generalized force F and its moment (DN, F) depend only on generalized velocities (quasi-velocities) and do not depend on the spatial position of the body. Then the system of equations (7), (15) on the manifold Rn × so(n) determines a closed system of dynamic equations of motion of a free n-dimensional rigid body subject to the external force F. This system can be separated from the kinematic part of the equations of motion on the manifold (6) and can be studied independently. 2. A More General Problem: Motion with a Following Force 2.1. Dynamic Part of the Equations of Motion. Consider the motion of a uniform dynamically symmetric (case (5)) rigid body whose “front face” (an (n − 1)-dimensional disk interacting with the medium that fills up the n-dimensional space) is subject to a resistance force S under the conditions of quasi-stationarity. Let (v, α, β1 , . . . , βn−2 ) be the (generalized) spherical coordinates of the velocity vector of a certain characteristic point D of the body (D is the center of the (n − 1)-dimensional disk on the axis of the dynamical symmetry of the body). Let Ω be the angular velocity tensor of the body and Dx1 . . . xn the coordinate system fixed to the body, with the symmetry axis CD coinciding with the axis Dx1 551

(C is the center of mass), and the axes Dx2 , Dx3 , . . . , Dxn lying on the hyperplane of the disk. Let I1 , I2 , I3 = I2 , . . . , In = I2 , m be the inertial and mass characteristics of the body. We assume the following representations with respect to the coordinates Dx1 . . . xn : DC = {−σ, 0, . . . , 0}, where

vD = viv (α, β1 , . . . , βn−2 ),

⎞ cos α ⎟ ⎜ sin α cos β1 ⎟ ⎜ ⎟ ⎜ sin α sin β1 cos β2 ⎟ ⎜ iv (α, β1 , . . . , βn−2 ) = ⎜ ⎟ . . . ⎟ ⎜ ⎝sin α sin β1 . . . sin βn−3 cos βn−2 ⎠ sin α sin β1 . . . sin βn−2

(17)

⎛

(18)

is the unit vector on the axis of the vector v. Moreover, in case (5), we adopt the following representation for the function characterizing the action of the medium on the n-dimensional body: S = {−S, 0, . . . , 0},

(19)

which means that F = S. Then we can write out the part of the dynamic equations of motion (also in the case of analytic Chaplygin functions [25]; see below) that describes the motion of the center of mass and corresponds to Rn , with no tangential forces exerted by the medium on the (n − 1)-dimensional disk. Thus, for n = 5, this system takes 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 S 2 + ω4 v sin α sin β1 sin β2 sin β3 + σ(ω10 + ω92 + ω72 + ω42 ) = − , m

(20)

v˙ sin α cos β1 + α˙ v cos α cos β1 − β˙1 v sin α sin β1 + 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, (21) v˙ sin α sin β1 cos β2 + α˙ v 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 + 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,

(22)

v˙ sin α sin β1 sin β2 cos β3 + α˙ v 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 − 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,

(23)

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,

(24)

where S = s(α)v 2 ,

σ = CD,

v > 0.

(25)

Further, the auxiliary matrix (14) for calculating the moment of the resistance force takes the form 0 x2N . . . xnN , (26) −S 0 ... 0 552

and then we can write out the part of the dynamic equations of motion describing the motion of the body about its center of mass and corresponding to the Lie algebra so(n). For n = 5, this system has the form (λ4 + λ5 ) ω˙ 1 + (λ4 − λ5 )(ω4 ω7 + ω3 ω6 + ω2 ω5 ) = 0,

(27)

(λ3 + λ5 ) ω˙ 2 + (λ5 − λ3 )(ω1 ω5 − ω3 ω8 − ω4 ω9 ) = 0,

(28)

(λ2 + λ5 ) ω˙ 3 + (λ2 − λ5 )(ω4 ω10 − ω2 ω8 − ω1 ω6 ) = 0,

(29)

Ω s(α)v 2 , (λ1 + λ5 ) ω˙ 4 + (λ5 − λ1 )(ω3 ω10 + ω2 ω9 + ω1 ω7 ) = −x5N α, β1 , β2 , β3 , v

(30)

(λ3 + λ4 ) ω˙ 5 + (λ3 − λ4 )(ω7 ω9 + ω6 ω8 + ω1 ω2 ) = 0,

(31)

(λ2 + λ4 ) ω˙ 6 + (λ4 − λ2 )(ω5 ω8 − ω7 ω10 − ω1 ω3 ) = 0,

(32)

Ω s(α)v 2 , (λ1 + λ4 ) ω˙ 7 + (λ1 − λ4 )(ω1 ω4 − ω6 ω10 − ω5 ω9 ) = x4N α, β1 , β2 , β3 , v (λ2 + λ3 ) ω˙ 8 + (λ2 − λ3 )(ω9 ω10 + ω5 ω6 + ω2 ω3 ) = 0,

Ω s(α)v 2 , (λ1 + λ3 ) ω˙ 9 + (λ3 − λ1 )(ω8 ω10 − ω5 ω7 − ω2 ω4 ) = −x3N α, β1 , β2 , β3 , v Ω s(α)v 2 . (λ1 + λ2 ) ω˙ 10 + (λ1 − λ2 )(ω8 ω9 + ω6 ω7 + ω3 ω4 ) = x2N α, β1 , β2 , β3 , v

(33) (34) (35) (36)

Thus, the phase space of the 15th-order system (20)–(24), (27)–(36) is the direct product of a 5-dimensional manifold and the Lie algebra so(5): R1 × S4 × so(5).

(37)

In the general case, the phase space has the form R1 × Sn−1 × so(n).

(38)

2.2. Implications of the Dynamic Symmetry. First, we note that due to the dynamic symmetry I2 = · · · = In ,

(39)

system (7) admits the cyclic first integrals (n − 1)(n − 2) . 2 Here, k1 = 1, . . . , ks are mutually distinct s integers from the set W1 = {1, 2, . . . , n(n − 1)/2}. Consider the 0th level of the first integrals (40): ωk1 ≡ ωk01 = const, . . . , ωks ≡ ωk0s = const,

s=

ωk01 = · · · = ωk0s = 0.

(40)

(41)

In particular, system (20)–(24), (27)–(36) admits the first integrals ω1 ≡ ω10 ,

ω2 ≡ ω20 ,

ω3 ≡ ω30 ,

ω5 ≡ ω50 ,

ω6 ≡ ω60 ,

ω8 ≡ ω80

(42)

considered on their 0th levels: ω10 = ω20 = ω30 = ω50 = ω60 = ω80 = 0. There remain

(43)

n(n − 1) (n − 1)(n − 2) − =n−1 2 2 of the tensor Ω (here, r1 , . . . , rp are the remaining p integers from W1

p= nonzero components ωr1 , . . . , ωrp distinct from k1 , . . . , ks ).

553

2.3. A Nonintegrable Constraint and the Choice of the Following Force. Consider a more general problem of a body moving in the presence of a following force T lying on the straight line CD = Dx1 and ensuring that during the motion we always have (see also [19, 28]) v ≡ const.

(44)

Then F1 in system (7), (15) (or, in particular, for n = 5 in system (20)–(24), (27)–(36)) is replaced by T − s(α)v 2 ,

σ = DC.

(45)

By choosing a suitable value of the following force T , one can formally ensure that relation (44) holds during the motion. Indeed, with the help of (7), (15), we obtain the following formal expression of T for cos α = 0, n > 2: mσ Ω sin α 2 2 2 , Γv α, β1 , . . . , βn−2 , T = Tv (α, β1 , . . . , βn−2 , Ω) = mσ(ωr1 + · · · + ωrp ) + s(α)v 1 − (n − 2)I2 cos α v (46) where

Ω = |rN | = rN , iN (β1 , . . . , βn−2 ) Γv α, β1 , . . . , βn−2 , v n π Ω = 0 · cos + xsN α, β1 , . . . , βn−2 , isN (β1 , . . . , βn−2 ). (47) 2 v s=2

Here, isN (β1 , . . . , βn−2 ), s = 1, . . . , n (i1N (β1 , . . . , βn−2 ) ≡ 0) are the components of the unit vector along the axis of the vector rN = {0, x2N , . . . , xnN } on the (n − 2)-dimensional sphere Sn−2 {β1 , . . . , βn−2 } defined by the relation α = π/2 as an equatorial section of the corresponding (n − 1)-dimensional sphere Sn−1 {α, β1 , . . . , βn−2 } (defined by (44)). Thus, we have (see (18)) ⎛ ⎞ ⎞ ⎛ 0 0 ⎜ i2N (β1 , . . . , βn−2 ) ⎟ ⎜ ⎟ cos β1 ⎜ ⎟ ⎟ ⎜ ⎜ i3N (β1 , . . . , βn−2 ) ⎟ ⎜ ⎟ sin β cos β 1 2 ⎟ = iv π , β1 , . . . , βn−2 . ⎟=⎜ iN (β1 , . . . , βn−2 ) = ⎜ ⎜ ⎟ ⎟ ⎜ ... ... 2 ⎜ ⎟ ⎟ ⎜ ⎝in−1N (β1 , . . . , βn−2 )⎠ ⎝sin β1 . . . sin βn−3 cos βn−2 ⎠ sin β1 . . . sin βn−2 inN (β1 , . . . , βn−2 ) (48) For the derivation of (46), we have used the conditions (40)–(44). 2.4. Reductions in the System. This procedure can be viewed from two standpoints. On one hand, the system has been transformed due to the presence of the following force (control), which allows us to consider the desired class of motions (44). On the other hand, all this can be viewed as a procedure for the reduction of the system’s order. Indeed, system (7), (15), after the said operations (ensuring (44), (40), (41)), generates an independent system of order n(n + 1) (n − 1)(n − 2) − − 1 = 2 (n − 1) (49) 2 2 or, in particular, for n = 5, system (20)–(24), (27)–(36) generates an independent system of order 8, namely, the system

554

αv ˙ cos α cos β1 − β˙ 1 v sin α sin β1 + ω10 v cos α − σ ω˙ 10 = 0,

(50)

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,

(51)

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,

(52)

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, Ω 3I2 ω˙ 4 = −x5N α, β1 , β2 , β3 , s(α)v 2 , 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 3I2 ω˙ 10 = x2N α, β1 , β2 , β3 , s(α)v 2 , v

(53) (54) (55) (56) (57)

which, in addition to the constant parameters specified above, contains the parameter v. System (50)–(57) is equivalent to 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,

(58)

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 t} = 0,

(59)

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, 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 s(α), x5N α, β1 , β2 , β3 , ω˙ 4 = − 3I2 v v2 Ω x4N α, β1 , β2 , β3 , s(α), ω˙ 7 = 3I2 v v2 Ω s(α), x3N α, β1 , β2 , β3 , ω˙ 9 = − 3I2 v v2 Ω x2N α, β1 , β2 , β3 , s(α). ω˙ 10 = 3I2 v

(60) (61) (62) (63) (64) (65)

2.5. New Quasi-Velocities in the System. In order to introduce new quasi-velocities in system (7), (15), we transform ωr1 , . . . , ωrn−1 with the help of a composition of the following n − 2 rotations: ⎞ ⎞ ⎛ ωr 1 z1 ⎟ ⎜ ⎜ z2 ⎟ ⎟ = Tn−2,n−1 (−β1 ) ◦ Tn−3,n−2 (−β2 ) ◦ · · · ◦ T1,2 (−βn−2 )⎜ ωr2 ⎟, ⎜ ⎝ ... ⎠ ⎝ ... ⎠ zn−1 ωrn−1 ⎛

(66)

where Tk,k+1 (β), k = 1, . . . , n − 2, is the matrix obtaine from the identity matrix by the insertion of a second-order minor Mk,k+1 , 555

⎛

Tk,k+1

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

1

0 0 ⎜ ... 0 ⎜0 ⎜ = ⎜0 0 Mk,k+1 ⎜ ⎝0 0 0 0 0 0

0

0

⎞

⎟ 0 0⎟ ⎟ 0 0⎟ , ⎟ .. . 0⎠ 0 1

(67)

Mk,k+1 =

mk,k = mk+1,k+1 = cos β,

mk+1,k = −mk,k+1 = sin β.

In particular, for n = 5, we introduce new quasi-velocities in system (20)–(24), (27)–(36). To this end, we transform ω4 , ω7 , ω9 , ω10 with the help of a composition of the following three rotations: ⎛ ⎞ ⎛ ⎞ z1 ω4 ⎜z2 ⎟ ⎜ ω7 ⎟ ⎜ ⎟ = T3,4 (−β1 ) ◦ T2,3 (−β2 ) ◦ T1,2 (−β3 ) ⎜ ⎟ , (68) ⎝z3 ⎠ ⎝ ω9 ⎠ z4 ω10 where

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

1 ⎜0 T3,4 (β) = ⎜ ⎝0 0

0 0 1 0

⎞ 0 0⎟ ⎟. 0⎠ 1

Thus, in particular, for system (20)–(24), (27)–(36) we have the following relations: 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 ,

(69)

z4 = [(ω7 cos β3 − ω4 sin β3 ) sin β2 − ω9 cos β2 ] sin β1 + ω10 cos β1 . 2.6. Systems of Normal Form. Relations (58)–(65) show that on the manifold π (70) O1 = (α, β1 , β2 , β3 , ω4 , ω7 , ω9 , ω10 ) ∈ R8 : α = k, β1 = πl, β2 = πm, k, l, m ∈ Z , 2 a solution of the system with respect to α, ˙ β˙ 1 , β˙ 2 , β˙ 3 may not be unique. Thus, formally, the uniqueness theorems do not hold on the manifold (70). Moreover, for k even and any l, m, there is uncertainty due to the degeneration of the spherical coordinates (v, α, β1 , β2 , β3 ), while for k odd, the uniqueness theorem is obviously violated, since the first equation in (58) degenerates. It follows that it is outside and only outside the manifold (70) that system (58)–(65) is equivalent to the system σv s(α) Ω Γv α, β1 , β2 , β3 , , (71) α˙ = −z4 + 3I2 cos α v v2 Ω cos α s(α)Γv α, β1 , β2 , β3 , z˙4 = − (z12 + z22 + z32 ) 3I2 v sin α Ω Ω Ω σv s(α) −z3 Δv,1 α, β1 , β2 , β3 , z2 Δv,2 α, β1 , β2 , β3 , − z1 Δv,3 α, β1 , β2 , β3 , , (72) + 3I2 sin α v v v 556

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

+

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

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

(73)

(74)

z˙1 = z1 z4

where

Δv,1 α, β1 , β2 , β3 , Δv,2 α, β1 , β2 , β3 , Δv,3 α, β1 , β2 , β3 ,

Ω π = rN , iN β1 + , β2 , β3 , v 2 Ω π π , β2 + , β3 , = rN , iN v 2 2 π π Ω π , , , β3 + = rN , iN v 2 2 2

(75) (76) (77) (78)

(79)

and the function Γv (α, β1 , β2 , β3 , Ω/v) is represented in the form (47). On the manifold O1 = (α, β1 , . . . , βn−2 , ωr1 , . . . , ωrn−1 ) ∈ R2(n−1) : π (80) α = k, β1 = πl1 , . . . , βn−3 = πln−3 , k, l1 , . . . , ln−3 ∈ Z 2 a solution of the system with respect to α, ˙ β˙ 1 , . . . , β˙ n−2 may not be unique. Thus, formally, the uniqueness theorem does not hold on the manifold (80). Moreover, for k even and any l1 , . . . , ln−3 , there is uncertainty due to the degeneration of the spherical coordinates (v, α, β1 , . . . , βn−2 ), while for k odd, the uniqueness theorem is obviously violated, since one of the equations degenerates. 557

It follows that it is outside and only outside the manifold (80) that system (7), (15) can be reduced to the system (n > 2) σv Ω s(α) Γv α, β1 , . . . , βn−2 , α˙ = −zn−1 + , (n − 2)I2 cos α v v2 Ω cos α 2 − (z12 + · · · + zn−2 s(α)Γv α, β1 , . . . , βn−2 , ) z˙n−1 = (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

(81)

(82)

s=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 cos 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

z˙n−2 = zn−2 zn−1

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

(83)

z˙n−3 = zn−3 zn−1

(84)

... Ω v2 s(α)Δv,n−2 α, β1 , . . . , βn−2 , (n − 2)I2 v

z˙1 = β˙ n−2 (−ωr1 sin βn−2 + ωr2 cos βn−2 ) + (−1)n cos α = z1 sin α

n−2 (−1)s+1 zn−s

cos βs−1 sin β1 . . . sin βs−1 s=1 n−1 Ω s(α) cos βs−1 σv n+1 s (−1) Δv,n−2 α, β1 , . . . , βn−2 , (−1) zn+1−s + (n − 2)I2 cos α v sin β1 . . . sin βs−1 s=2 Ω v2 , (85) s(α)Δv,n−2 α, β1 , . . . , βn−2 , + (−1)n (n − 2)I2 v σv Ω cos α s(α) ˙ + Δv,1 α, β1 , . . . , βn−2 , , β1 = zn−2 sin α (n − 2)I2 sin α v 558

(86)

β˙ 2 = −zn−3

σv Ω cos α s(α) , + Δv,2 α, β1 , β2 , β3 , sin α sin β1 (n − 2)I2 sin α sin β1 v

(87)

... cos α 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

β˙ n−2 = (−1)n+1 z1

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 ... π π Ω π = rN , iN , . . . , , βn−3 + , βn−2 , Δv,n−3 α, β1 , . . . , βn−2 , ], v 2 2 2 π π Ω π Δv,n−2 α, β1 , . . . , βn−2 , , , . . . , , βn−2 + = rN , iN v 2 2 2

(88)

(89)

and the function Γv (α, β1 , . . . , βn−2 , Ω/v) is represented in the form (47). Here and in what follows, the dependence on the groups of variables Ω α, β1 , . . . , βn−2 , v is understood as a composite dependence on zn−1 z1 α, β1 , . . . , βn−2 , , . . . , v v due to (68). 2.7. Remarks on the Distribution of Indices. The right-hand sides of system (81)–(88), after the common coefficient s(α) σv , (n − 2)I2 cos α are linear with respect to the quantities Δv,s (α, β1 , . . . , βn−2 , Ω/v), s = 1, . . . , n − 2, (always n − 2 in number). For instance, equation (82) (with the left-hand side z˙n−1 ) contains the functions (89) with all indices s from 1 to n − 2 (each index once), i.e., 1 2 3 4 . . . n − 2.

(90)

However, further in equations (83)–(85), the functions (89) appear in a different manner. For instance, the equation for z˙n−2 still contains the functions (89) with the indices (90), but the equation for z˙n−3 contains the functions with the indices 2 2 3 4 . . . n − 2,

(91)

i.e., the function Δv,2 (α, β1 , . . . , βn−2 , Ω/v) enters twice. What is the general distribution of the indices? The answer is given in Table 1. 559

Table 1. General distribution of indices of functions (89) Left-hand side of system (81)–(88) z˙n−2 z˙n−3 z˙n−4 z˙n−5 ... z˙1 Thus, the first-order minor

Distribution of indices s of functions (89) 1 2 3 4 ... 2 2 3 4 ... 3 3 3 4 ... 4 4 4 4 ... ... ... ... ... ... n − 2 n − 2 n − 2 n − 2 ...

n−2 n−2 n−2 n−2 ... n−2

1

in the top left corner of Table 1 corresponds to the case n = 3 and shows that the dynamic equations contain only the functions (89) (for s = 1). The second-order minor 1 2 2 2 at the same place corresponds to the case n = 4 and shows that the dynamic equations contain the functions (89) (for s = 1, 2). The third-order minor ⎛ ⎞ 1 2 3 ⎝2 2 3⎠ 3 3 3 at the same place corresponds to the case n = 5 and shows that the dynamic equations (71)–(78) contain the functions (89) (for s = 1, 2, 3), etc. 2.8. Violation of the Uniqueness Theorem. The uniqueness theorem for system (7), (15) on the manifold (80) with k odd is violated in the following sense: almost any point of the manifold (80) for k odd is crossed by a nonsingular trajectory of system (7), (15), forming a right angle with the manifold (80), and there is also a phase trajectory that coincides with the said point at all times. Physically, these trajectories are different, since they correspond to different values of the following force. Let us prove this statement. It has been shown above that in order to maintain the constraint (44), it is necessary to choose T of the form (46) for cos α = 0. Let Ω Ω s(α) = L β1 , . . . , βn−2 , . (92) Γv α, β1 , . . . , βn−2 , lim v v α→π/2 cos α Note that |L| < +∞, if and only if

∂

Ω Γv α, β1 , . . . , βn−2 , s(α)

< +∞. lim

v α→π/2 ∂α For α = π/2, the required value of the following force is found from the relation π mσLv 2 2 , β1 , . . . , βn−2 , Ω = mσ(ωr21 + · · · + ωn−1 )− , T = Tv 2 (n − 2)I2 where ωr1 , . . . , ωn−1 are arbitrary. 560

(93)

n > 2,

(94)

On the other hand, to maintain rotation about some point W with the help of the following force, it should be chosen of the form π mv 2 , (95) , β1 , . . . , βn−2 , Ω = T = Tv 2 R0 where R0 is the distance CW . Relations (94) and (95), in general, determine different values of the following force T for almost all points of the manifold (80), which proves the above statement. 3. Integrable Systems on the Tangent Bundle of a Finite-Dimensional Sphere 3.1. Reduced System. By analogy with choosing the analytic Chaplygin functions [25], we utilize (48) and choose the dynamic functions s, x2N , . . . , xnN in the form s(α) = B cos α, rN = R(α)iN , R(α) = A sin α,

A, B > 0,

(96)

which allows us to claim that in the system under consideration, the moment of nonconservative forces does not depend on the angular velocity (it depends only on the angles α, β1 , . . . , βn−2 ). Moreover, the functions Ω Ω Γv α, β1 , . . . , βn−2 , , Δv,s α, β1 , . . . , βn−2 , , s = 1, . . . , n − 2, v v in system (81)–(88) take the form Ω = R(α) = A sin α, Γv α, β1 , . . . , βn−2 , v Ω ≡ 0, s = 1, . . . , n − 2. Δv,s α, β1 , . . . , βn−2 , v

(97)

Then, due to the nonintegrable constraint (44), outside and only outside the manifold (80) the dynamic part of the equations of motion (system (81)–(88)) turns into the analytic system α˙ = −zn−1 + z˙n−1 =

σABv sin α, (n − 2)I2

(98)

ABv 2 cos α 2 , sin α cos α − (z12 + · · · + zn−2 ) (n − 2)I2 sin α

(99)

z˙n−2 = zn−2 zn−1

cos α cos β1 cos α 2 ) , + (z12 + · · · + zn−3 sin α sin α sin β1

(100)

z˙n−3 = zn−3 zn−1

cos α 1 cos β2 cos α cos α cos β1 2 − zn−3 zn−2 − (z12 + · · · + zn−4 ) , sin α sin α sin β1 sin α sin β1 sin β2

(101)

... cos α z˙1 = z1 sin α

n−2 (−1)s+1 zn−s s=1

cos α , β˙ 1 = zn−2 sin α cos α , β˙ 2 = −zn−3 sin α sin β1 ...

cos βs−1 , sin β1 . . . sin βs−1

(102) (103) (104)

561

cos α , sin α sin β1 . . . sin βn−4

β˙ n−3 = (−1)n z2

β˙ n−2 = (−1)n+1 z1

(105)

cos α . sin α sin β1 . . . sin βn−3

(106)

Introducing dimensionless parameters and differentiation by zk → n0 vzk , k = 1, . . . , n − 1, n20 =

AB (n > 2), (n − 2)I2

b = σn0 ,

· = n0 v ,

(107)

we reduce system (98)–(106) to α = −zn−1 + b sin α,

(108)

cos α , sin α cos α cos β1 cos α 2 + (z12 + · · · + zn−3 ) , sin α sin α sin β1

2 = sin α cos α − (z12 + · · · + zn−2 ) zn−1

(109)

= zn−2 zn−1 zn−2

(110)

zn−3 = zn−3 zn−1

... z1

cos α = z1 sin α

cos α 1 cos β2 cos α cos α cos β1 2 − (z12 + · · · + zn−4 ) , − zn−3 zn−2 sin α sin α sin β1 sin α sin β1 sin β2

n−2 (−1)s+1 zn−s s=1


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

= (−1)n z2 βn−3

(112) (113) (114)


= (−1)n+1 z1 βn−2

(111)

(115)


(116)

In particular, for n = 5, we obtain the following 8th-order system: α = −z4 + b sin α, z4 = sin α cos α − (z12 + z22 + z32 )

cos α , sin α

(118)

z3 = z3 z4

cos α cos β1 cos α + (z12 + z22 ) , sin α sin α sin β1

(119)

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

(120)

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

(121)

β1 = z3 562

(117)

cos α , sin α

(122)

β2 = −z2 β3 = z1

cos α , sin α sin β1

cos α . sin α sin β1 sin β2

(123) (124)

We see that within the 2(n − 1)th-order system (108)–(116), which can be considered on 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 }, there appears an independent system (108)–(115) of order 2n − 3 on its own (2n − 3)-dimensional manifold (in particular, within the 8th-order system (117)–(124), which can be considered on the tangent bundle T∗ S4 of the four-dimensional sphere S4 , there appears an independent system (117)–(123) of order 7 on its own 7-dimensional manifold). In the general case, the following result holds. Theorem 3.1. Under the conditions (44), (40), (41), system (7), (15) is reduced to the dynamical system (81)–(88) on 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 }. 3.2. General Remarks about Integrability. For the 2(n − 1)th-order system (108)–(116) to be completely integrable, in general, it is necessary to have its 2n − 3 independent first integrals (in particular, for the complete integrability of the 8th-order system (117)–(124), in general, we should know its 7 independent first integrals). However, the systems under consideration have symmetries which allow us to reduce the sufficient number of first integrals to n (in particular, to 5). 3.2.1. The system in the absence of forces. First, consider system (117)–(124) on the tangent bundle T∗ S4 {z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 } of the 4-dimensional S4 {α, β1 , β2 , β3 }. From this, we obtain a conservative system. Moreover, assume that the function (47) identically vanishes (in particular, b = 0, and the coefficient sin α cos α in equation (118) is absent). The system under consideration takes the form α = −z4 ,

(125)

z4 = −(z12 + z22 + z32 )

cos α , sin α

(126)

z3 = z3 z4


(127)

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

(128)

z1 = 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

(129)

cos α , sin α cos α , β2 = −z2 sin α sin β1

β1 = z3

β3 = z1


(130) (131) (132)

System (125)–(132) describes motion of a rigid body in the absence of external forces. 563

Theorem 3.2. System (125)–(132) admits five independent analytic first integrals of the form Φ1 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = z12 + z22 + z32 + z42 = C1 = const,

(133)

Φ2 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) =

z12 + z22 + z32 sin α = C2 = const,

(134)

Φ3 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) =

z12 + z22 sin α sin β1 = C3 = const,

(135)

Φ4 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = z1 sin α sin β1 sin β2 = C4 = const,

(136)

Φ5 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C5 = const.

(137)

The first four first integrals (133)–(136) show that due to the absence of external forces, four components (nonzero, in general) of the angular velocity tensor of the 5-dimensional body are preserved: ω4 ≡ ω40 = const,

ω7 ≡ ω70 = const,

ω9 ≡ ω90 = const,

0 ω10 ≡ ω10 = const.

(138)

In particular, the existence of the first integral (133) is due to the relation 2 ≡ C12 = const. z12 + z22 + z32 + z42 = ω42 + ω72 + ω92 + ω10

(139)

The fifth first integal (137) can be interpreted in a kinematic sense: it “attaches” the equation for β3 and can be found from the following quadrature: dβ3 z1 1 =− , (140) dβ2 z2 sin β2 and if, in addition, we take into account the levels of the first integrals (135), (136) and obtain the relation z1 C32 =± sin2 β2 − 1, (141) z2 C42 then we can write the quadrature (140) in the form du β3 = ± C32 (1 − u2 ) − 1 − 2 C 4

C32 2 u C42

,

u = cos β2 .

(142)

C5 = const,

(143)

Calculations show that β3 + C5 = ± arctan

cos β2 C32 C42

2

,

sin β2 − 1

which allows us to obtain the first integral (137). From (143), we obtain the following invariant relation: tan2 (β3 + C5 ) =

C42 . (C32 − C42 ) tan2 β2 − C42

(144)

Now we can restate Theorem 3.2 as follows. Theorem 3.3. System (125)–(132) admits five independent first integrals of the form Ψ1 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) =

Φ21 z2 + z2 + z2 + z2 = 1 2 2 2 3 2 4 = C1 = const, Φ2 z1 + z2 + z3 sin α

Ψ2 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C2 = const, z12 + z22 Φ3 Ψ3 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = = = C3 = const, Φ4 z1 sin β2 564

(145) (146) (147)

z 2 + z22 + z32 Φ2 Ψ4 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = = 21 = C4 = const, 2 Φ3 z1 + z2 sin β1

(148)

Ψ5 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C5 = const.

(149)

The fifth first integral (149) also has kinematic sense and “attaches” the equation for β3 , while the functions Ψ2 , Ψ5 can be chosen equal to Φ2 , Φ5 , respectively. Theorem 3.3 (in contrast to Theorem 3.2) does not specify smoothness of the first integrals. Namely, the points at which the denominators (or numerators and denominators simultaneously) of the first integrals (145)–(149) vanish are singular points of the integrals regarded as functions. Moreover, they can even be discontinuous, in general. In view of Theorem 3.3, the transformed set of the first integrals (145)–(149) of system (125)–(132) (without a force field) is still a set of its first integrals. For the complete integrability of the 8th-order system (125)–(132), one has to know its seven independent first integrals. However, after the transformation of the variables ⎛ ⎞ ⎛ ⎞ w4 z4 ⎜w3 ⎟ ⎜z3 ⎟ ⎜ ⎟ → ⎜ ⎟ , w4 = z4 , w3 = z 2 + z 2 + z 2 , w2 = z2 , w1 = z3 , (150) 1 2 3 ⎝w2 ⎠ ⎝z2 ⎠ z1 z12 + z22 z1 w1 system (125)–(132) splits into the equations α = −w4 ,

(151)

cos α , sin α cos α w3 = w3 w4 , sin α

w4 = −w32

(152) (153)

w2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 )

1 + w22 cos β2 , w2 sin β2

β2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ), w1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) β1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(154) 1 + w12 cos β1 , w1 sin β1

β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ), where

cos α , sin α cos α , d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = −Z2 (w4 , w3 , w2 , w1 ) sin α sin β1 cos α , d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = Z1 (w4 , w3 , w2 , w1 ) sin α sin β1 sin β2

(155) (156)

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

(157)

and zk = Zk (w4 , w3 , w2 , w1 ),

k = 1, 2, 3,

(158)

are the functions defined by (150). We see that the 8th-order system (151)–(156) splits into independent lower order subsystems: (151)–(153) of order 3 and systems (154), (155) (after changing the independent variable) of order 2. 565

Thus, for the complete integrability of system (151)–(156), it suffices to find two independent first integrals of system (151)–(153), one integral for system (154), one integral for system (155), and an additional first integral that “attaches” equation (156) (i.e., five integrals altogether ). Remark 3.1. Let us write out the first integrals (145)–(149) in the variables w1 , w2 , w3 , w4 defined by (150). We have Θ1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) =

w32 + w42 = C1 = const, w3 sin α

(159)

Θ2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = w3 sin α = C2 = const, 1 + w22 = C3 = const, Θ3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = sin β2 1 + w12 = C4 = const, Θ4 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = sin β1

(161)

Θ5 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = C5 = const.

(163)

(160)

(162)

Thus, two independent first integrals (159), (160) are sufficient for the integration of system (151)–(153); the first integrals (161), (162) are sufficient for the integration of two independent first-order equations 1 + ws2 cos βs dws = , s = 1, 2, (164) dβs ws sin βs which (after changing the independent variable) are equivalent to systems (154), (155), respectively; finally, the first integral (163) is sufficient for “attaching” equation (156). Thus, we have proved the following result. Theorem 3.4. The 8th-order system (125)–(132) possesses sufficiently many (five) independent first integrals. 3.2.2. System with a conservative field of forces. Consider system (117)–(124) with b = 0. We thus have a conservative system. The field of forces is characterized by the term sin α cos α in equation (118) (in contrast to system (125)–(132)). The system under consideration takes the form α = −z4 ,

(165)

z4 = sin α cos α − (z12 + z22 + z32 )

(166)

z3 = z3 z4

cos α cos β1 cos α , + (z12 + z22 ) sin α sin α sin β1

(167)

z2 = z2 z4


(168)

z1 = z1 z4


(169)


β1 = z3

β3 = z1 566

cos α , sin α


(170) (171) (172)

Thus, system (165)–(172) describes motion of a rigid body in a conservative field of external forces. Theorem 3.5. System (165)–(172) admits five independent analytic first integrals of the form Φ1 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = z12 + z22 + z32 + z42 + sin2 α = C1 = const, Φ2 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = z12 + z22 + z32 sin α = C2 = const, Φ3 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) =


(173) (174) (175)

Φ4 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = z1 sin α sin β1 sin β2 = C4 = const,

(176)

Φ5 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C5 = const.

(177)

The first integral (173) is the integral of total energy. The fifth first integral (177) has a kinematic sense; it “attaches” the equation for β3 and has been found above. Let us restate Theorem 3.5. Theorem 3.6. System (165)–(172) admits five independent first integrals of the form Ψ1 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) =

Φ1 z 2 + z 2 + z32 + z42 + sin2 α = C1 = const, = 1 22 2 2 Φ2 z1 + z2 + z3 sin α

Ψ2 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C2 = const, z12 + z22 Φ3 = = C3 = const, Ψ3 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = Φ4 z1 sin β2 z 2 + z22 + z32 Φ2 = 21 = C4 = const, Ψ4 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = 2 Φ3 z1 + z2 sin β1 Ψ5 (z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 ) = C5 = const.

(178) (179) (180) (181) (182)

The functions Ψ2 , Ψ5 can be chosen equal to Φ2 , Φ5 , respectively. Theorem 3.6 (in contrast to Theorem 3.5) does not specify the smoothness of the first integrals. Thus, the points at which the denominators (or nominators and denominators simultaneously) of the first integrals (178)–(182) vanish are points of singularity of these integrals regarded as functions, which often happen to be discontinuous, in general. Due to Theorem 3.6, the transformed set of first integrals (178)–(182) of system (165)–(172) (with conservative forces) is still a set of its first integrals. For the complete integrability of the 8th-order system (165)–(172), in general, it is necessary to know its seven independent first integrals. However, after the transformation of the variables (150), system (165)–(172) splits into the equations α = −w4 ,

(183)

w4 = sin α cos α − w32 w3 = w3 w4

cos α , sin α

(184)

cos α , sin α

w2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) β2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(185) 1 + w22 cos β2 , w2 sin β2 (186) 567

w1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) β1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

1 + w12 cos β1 , w1 sin β1

β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(187) (188)

with the conditions (157). We see that the 8th-order system (183)–(188) splits into independent subsystems of lower order: system (183)–(185) is of order 3, and systems (186), (187) (after changing the independent variable) are of order 2. Thus for system (183)–(188) to be completely integrable, it suffices to find two independent first integrals of system (183)–(185), one integral of system(186), one integral of system (187), and an additional first integral that “attaches” equation (188) (five integrals altogether ). Remark 3.2. The first integrals (178)–(182) in the variables w1 , w2 , w3 , w4 defined in (150) take the form Θ1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) =

w32 + w42 + sin2 α = C1 = const, w3 sin α

(189)

Θ2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = w3 sin α = C2 = const, 1 + w22 = C3 = const, Θ3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = sin β2 1 + w12 = C4 = const, Θ4 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = sin β1

(190)

Θ5 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ) = C5 = const.

(193)

(191)

(192)

Thus, two independent first integrals (189), (190) are sufficient for the integration of system (183)–(185); the first integrals (191), (192) are sufficient for the integration of two independent first-order equations 1 + ws2 cos βs dws = , s = 1, 2, (194) dβs ws sin βs which, upon changing the independent variable, are equivalent to systems (186), (187), respectively; the first integral (193) is sufficient for “attaching” equation (188). Thus, we have established the following result. Theorem 3.7. System (165)–(172) of order 8 possesses sufficiently many (five) independent first integrals. 3.3. Complete List of Invariant Relations. Let us integrate the 8th-order system (117)–(124) (without simplifications, with all coefficients present). As above, for the complete integrability of the 8th-order system (117)–(124), it is necessary, in general, to know its seven independent first integrals. However, after the transformation of the variables (150), system (117)–(124) splits into the equations α = −w4 + b sin α, w4 = sin α cos α − w32 w3 = w3 w4

(195) cos α , sin α

cos α , sin α

1 + w22 cos β2 , w2 sin β2 β2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(196) (197)

w2 = d2 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 )

568

(198)

1 + w12 cos β1 , w1 sin β1 β1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(199)

β3 = d3 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 ),

(200)

w1 = d1 (w4 , w3 , w2 , w1 ; α, β1 , β2 , β3 )

with the conditions (157)). We see that the 8th-order system (195)–(200) splits into lower order independent subsystems: system (195)–(197) of order 3, and systems (198) and (199) (after changing the independent variable) of order 2. Thus, for the complete integration of system (195)–(200), it suffices to find two independent first integrals of system (195)–(197), one integral of system (198), one integral of system (199), and an additional first integral “attaching” equation (200) (five integrals altogether ). First, we associate the third-order system (195)–(197) with the nonautonomous second-order system sin α cos α − w32 cos α/ sin α dw4 = , dα −w4 + b sin α

dw3 w3 w4 cos α/ sin α = . dα −w4 + b sin α

(201)

Setting τ = sin α, we rewrite system (201) in algebraic form, τ − w32 /τ dw4 = , dτ −w4 + bτ

dw3 w3 w4 /τ = . dτ −w4 + bτ

(202)

w4 = u2 τ,

(203)

Next, introducing homogeneous variables, w3 = u1 τ, we reduce system (202) to τ

1 − u21 du2 + u2 = , dτ −u2 + b

τ

du1 u1 u2 + u1 = , dτ −u2 + b

(204)

which is equivalent to 1 − u21 + u22 − bu2 du1 2u1 u2 − bu1 du2 = , τ = . dτ −u2 + b dτ −u2 + b Let us associate the second-order system (205) with the nonautonomous first-order equation τ

1 − u21 + u22 − bu2 du2 = , du1 2u1 u2 − bu1 which can be easily reduced to the total differential 2 u2 + u21 − bu2 + 1 = 0. d u1

(205)

(206)

(207)

Thus, equation (206) has the first integral u22 + u21 − bu2 + 1 = C1 = const, u1

(208)

which, in former variables, can be written as Θ1 (w4 , w3 ; α) =

w42 + w32 − bw4 sin α + sin2 α = C1 = const. w3 sin α

(209)

Remark 3.3. For b = 0, the first integral (209) of system (195)–(197) coincides with the first integral (189) of system (183)–(185), but for b = 0, neither the numerator of the expression (209) nor its denominator are, separately, first integrals of system (195)–(197) (although, for b = 0, both the numerator and the denominator of (209) are first integrals of system (183)–(185)). 569

Now, let us find, in explicit form, the additional first integral of the third-order system (195)–(197). To this end, we first transform the invariant relation (208) for u1 = 0 as follows: b 2 C1 2 b2 + C12 u2 − + u1 − = − 1. (210) 2 2 4 We see that the parameters of this invariant relation must satisfy the condition b2 + C12 − 4 ≥ 0,

(211)

and the phase space of system (195)–(197) foliates into a family of surfaces defined by (210) in the coordinates u1 , u2 . Thus, due to (208), the first equation of system (205) takes the form τ

2(1 − bu2 + u22 ) − C1 U1 (C1 , u2 ) du2 = , dτ −u2 + b

(212)

where

1 (213) C1 ± C12 − 4(u22 − bu2 + 1) , 2 and the integration constant C1 is chosen from the condition (211). Therefore, the quadrature for finding the additional first integral of system (195)–(197) takes the form (b − u2 ) du2 dτ = . (214) 2 τ 2(1 − bu2 + u2 ) − C1 {C1 ± C12 − 4(u22 − bu2 + 1)}/2 U1 (C1 , u2 ) =

The left-hand side (to within an additive constant) is obviously equal to ln | sin α|.

(215)

If b (216) = r1 , b21 = b2 + C12 − 4, 2 then the right-hand side of (214) takes the form

b d(b21 − 4r12 ) 1

b21 − 4r12 dr1 1

± I1 , (217) − b = − ± 1 ln −

2 4 2

C1 (b21 − 4r12 ) ± C1 b21 − 4r12 (b21 − 4r12 ) ± C1 b21 − 4r12 u2 −

where

dr3

r3 = b21 − 4r12 .

(218) , b21 − r32 (r3 ± C1 ) When calculating the integral (218), we consider three possible cases: (I) b > 2.

√ 2

b − 4 + b21 − r32 1 C1

ln ±√ I1 = − √ r3 ± C1 2 b2 − 4

b2 − 4

√ 2

b − 4 − b21 − r32 C1

1

ln ∓√ + const. (219) + √ r3 ± C1 b2 − 4

2 b2 − 4

I1 =

(II) b < 2. I1 = √ (III) b = 2.

570

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

b21 − r32 + const. I1 = ∓ C1 (r3 ± C1 )

(220)

(221)

Going back to the variable r1 =

b r3 − , sin α 2

(222)

we finally obtain the following formulas for I1 . (I) b > 2.

√ 2

b − 4 ± 2r1 1 C1

ln I1 = − √ ±√ 2 b2 − 4 b21 − 4r12 ± C1 b2 − 4

√ 2

b − 4 ∓ 2r1

C 1 1

+ const. (223) √ ∓ ln

2 + √ 2 b2 − 4 b2 − 4

b1 − 4r12 ± C1

(II) b < 2.

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

(224)

(III) b = 2. 2r1 + const. (225) C1 ( b21 − 4r12 ± C1 ) Thus, we have found the additional first integral of the third-order system (195)–(197) and obtained a complete set of first integrals, which happen to be transcendental functions of their phase variables [2, 5, 6, 10, 12, 14, 17]. I1 = ∓

Remark 3.4. In the above expression of the first integral I1 , the constant C1 should be formally replaced by the left-hand side of the first integral (208). Then the additional first integral obtained above acquires the following structure: w4 w3 = C2 = const. , (226) Θ2 (w4 , w3 ; α) = ln | sin α| + G2 sin α, sin α sin α Thus, we have found two first integrals (209), (226) of the independent third-order system (195)–(197). It remains to indicate one integral for (198), one for (199), and an additional first integral “attaching” equation (200). The desired first integrals coincide with the first integrals (191)–(193), namely: 1 + w22 = C3 = const, (227) Θ3 (w2 ; β2 ) = sin β2 1 + w12 = C4 = const, (228) Θ4 (w1 ; β1 ) = sin β1 C4 cos β2 Θ5 (w2 , w1 ; β1 , β2 , β3 ) = β3 ± arctan = C5 = const, C32 sin2 β2 − C42

(229)

where C3 , C4 in the left-hand side of (229) should be replaced by the integrals (227), (228). Theorem 3.8. The 8th-order system (195)–(200) possesses sufficiently many (five) independent first integrals (209), (226), (227)–(229). Thus, in the case under consideration, the system of dynamic equations (20)–(24), (27)–(36) with the condition (96) admits 12 invariant relations: the analytic nonintegrable constraint (44); the cyclic first integrals (42), (43); the first integral (209); the first integral defined by (219)–(226), which is a transcendental function of the phase variables (in the sense of complex analysis) and has the form of a finite combination of elementary functions; and finally, the transcendental first integrals (227)–(229). 571

Theorem 3.9. System (20)–(24), (27)–(36) with the conditions (44), (96), (42), (43) admits 12 invariant relations (a complete set), five of which are transcendental functions in the sense of complex analysis. All these relations can be expressed in terms of finite combinations of elementary functions. 3.4. Structure of Equations on Tangent Bundles of Finite-Dimensional Spheres. Our examination of the complete system (117)–(124) on the tangent bundle T∗ S4 {z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 } of the four-dimensional sphere S4 {α, β1 , β2 , β3 } started with a simplified system (125)–(132) that describes the dynamics in the absence of external forces. Thus, the coefficients in system (125)–(132) have only a geometric meaning and are determined by our choice of the coordinates z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 on the tangent bundle. Our aim is to find out how the coefficients of the corresponding systems change in the process of induction, with the increase of the dimension n − 1 of the sphere Sn−1 {α, β1 , . . . , βn−2 }. In other words, what systems describe phase (geodesic) flows on 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 the coordinates zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ? We start with n = 2, in spite of the fact that here and in previous publications we have considered the structure of the corresponding equations for n ≤ 5. This will allow us to use induction for passing from n to n + 1 and “construct” similar systems of higher orders. Remark 3.5 (on analytic first integrals in the absence of forces). When constructing systems on 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 }, one utilizes the fact that the system has the following set of analytic first integrals: 2 = C1 = const, Φ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z12 + · · · + zn−1 Φ2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 z12 + · · · + zn−2 sin α = C2 = const,

Φ3 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 sin α sin β1 = C3 = const, z12 + · · · + zn−3

... Φn−2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

(230)

z12 + z22 sin α sin β1 . . . sin βn−4 = Cn−2 = const,

Φn−1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z1 sin α sin β1 . . . sin βn−3 = Cn−1 = const. The first integrals (230) show that due to the absence of external forces, n − 1 components (nonzero, in general) of the angular velocity tensor of the n-dimensional rigid body are preserved: ωr1 ≡ ωr01 = const,

...,

ωrn−1 ≡ ωr0n−1 = const.

(231)

In particular, the first integral Φ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = C1 in (230) is due to the relation 2 z12 + · · · + zn−1 = ωr21 + · · · + ωr2n−1 ≡ C12 = const.

(232)

At the same time, the first integrals (230) are functions of the components ωr1 , . . . , ωrn−1 . 3.4.1. Induction basis: n = 2. For n = 2, the following system specifies a geodesic flow on the twodimensional cylinder T∗ S1 {z1 ; α}, which is the tangent bundle of the one-dimensional sphere S1 {α}: α = −z1 ,

z1 = 0,

(233)

and, in view of Remark 3.5, there is a natural first integral z1 = C1 = const. 572

(234)

The equation α˙ = −z1 is a kinematic relation that defines the coordinates α, z1 in the phase space of system (233) (the tangent bundle T∗ S1 {z1 ; α}). 3.4.2. Transition in n: 2 → 3. When passing from n = 2 to n = 3, we change the index of the first variable, z1 → z2 , and introduce the new variable z1 . Moreover, in the system under construction, we underline new terms appearing as n is increased. Proposition 3.1. For n = 3, the following system defines a geodesic flow on the tangent bundle of the two-dimensional sphere T∗ S2 {z2 , z1 ; α, β1 } S2 {α, β1 }: α = −z2 ,

(235)

z2 = −z12

cos α , sin α

(236)

z1 = z1 z2

cos α , sin α

(237)

cos α , sin α and, in view of Remark 3.5, there exist the filrst integrals β1 = z1

(238)

z12 + z22 = C1 = const,

(239)

z1 sin α = C2 = const.

(240)

Indeed, due to (239), we have

z1 z1 + z2 z2 = 0, and thus, there is a function N1 (α, β1 , z1 , z2 ) such that z2 = −z1 N1 (α, β1 , z1 , z2 ),

z1 = z2 N1 (α, β1 , z1 , z2 ),

and (240) ensures that (see (235)–(238)) z1 sin α + z1 α cos α = z2 N1 (α, β1 , z1 , z2 ) sin α − z1 z2 cos α = 0, which implies that N1 (α, β1 , z1 , z2 ) = z1

cos α , sin α

as required. Equations (235), (238) are kinematic relations that define the coordinates α, β1 , z1 , z2 in the phase space of system (235)–(238) (the tangent bundle T∗ S2 {z2 , z1 ; α, β1 }). 3.4.3. Transition in n: 3 → 4. When passing from n = 3 to n = 4, we change the subscripts of the first two variables, z z2 → 3 , z1 z2 and inroduce a new z1 . We also underline the new terms that appear as n is increased. Proposition 3.2. For n = 4, the following system defines a geodesic flow on the tangent bundle T∗ S3 {z3 , z2 , z1 ; α, β1 , β2 } of the three-dimensional sphere S3 {α, β1 , β2 }: α = −z3 , z3 = −(z22 + z12 )


(242) 573

z2 = z2 z3

cos α cos α cos β1 + z12 , sin α sin α sin β1

(243)

z1 = z1 z3

cos α cos α cos β1 − z1 z2 , sin α sin α sin β1

(244)

cos α , sin α cos α β2 = −z1 , sin α sin β1 β1 = z2

(245) (246)

and, in view of Remark 3.5. there exist the first integrals z12 + z22 + z32 = C1 = const, z12 + z22 sin α = C2 = const,

(248)

z1 sin α sin β1 = C3 = const.

(249)

(247)

Indeed, using (247), (248) and arguing as in the proof of Proposition 3.1, we find the underlined coefficient in equation (242) and can now claim that equations (243) and (244) can be written in the form: cos α cos α + z12 N2 (α, β1 , β2 , z1 , z2 , z3 ), z2 = z2 z3 sin α sin α (250) cos α cos α z1 = z1 z3 − z1 z2 N2 (α, β1 , β2 , z1 , z2 , z3 ). sin α sin α Further, due to (249), (241)–(246), we have z1 sin α sin β1 + z1 α cos α sin β1 + z1 β1 sin α cos β1 = z1 z2 cos α[cos β1 − N2 (α, β1 , β2 , z1 , z2 , z3 ) sin β1 ] = 0, which implies that N2 (α, β1 , β2 , z1 , z2 , z3 ) =

cos β1 , sin β1

as required. Equations (241), (245), (246) are kinematic relations defining the coordinates α, β1 , β2 , z1 , z2 , z3 in the phase space of system (241)–(246) (the tangent bundle T∗ S3 {z3 , z2 , z1 ; α, β1 , β2 }). 3.4.4. Transition in n: 4 → 5. When passing from n = 4 to n = 5, we change the subscripts of the first three variables, ⎛ ⎞ ⎛ ⎞ z4 z3 ⎝z2 ⎠ → ⎝z3 ⎠ , z1 z2 and introduce a new z1 . In the system under construction, new terms that appear with the increase of n are underlined. Proposition 3.3. For n = 5, the following system defines a geodesic flow on the tangent bundle T∗ S4 {z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 } of the four-dimensional sphere S4 {α, β1 , β2 , β3 }: α = −z4 ,


(252)


(253)

z4 = −(z32 + z22 + z12 ) z3 = z3 z4 574

z2 = z2 z4


(254)

z1 = z1 z4


(255)

cos α , sin α cos α β2 = −z2 , sin α sin β1

β1 = z3

β3 = z1

(256) (257)

cos α , sin α sin β1 sin β2

(258)

and, due to Remark 3.5, there exist the first integrals z12 + z22 + z32 + z42 = C1 = const, z12 + z22 + z32 sin α = C2 = const,


z1 sin α sin β1 sin β2 = C4 = const.

(259) (260) (261) (262)

Using (259)–(261) and arguing as in the proof of Propositions 3.1, 3.2, we find the underlined coefficients in equations (252), (253) and can claim that equations (254) and (255) can be written as follows: z2 = z2 z4 z1

cos α cos α cos β1 cos α − z2 z3 N3 (α, β1 , β2 , β3 , z1 , z2 , z3 , z4 ), − z12 sin α sin α sin β1 sin α

cos α cos α cos β1 cos α = z1 z4 + z1 z2 − z1 z3 N3 (α, β1 , β2 , β3 , z1 , z2 , z3 , z4 ). sin α sin α sin β1 sin α

(263)

Further, due to (262), (251)–(258), we have z1 sin α sin β1 sin β2 + z1 α cos α sin β1 sin β2 + z1 β1 sin α cos β1 sin β2 + z1 β2 sin α sin β1 cos β2 = z1 z2 cos α[N3 (α, β1 , β2 , β3 , z1 , z2 , z3 , z4 ) sin β1 sin β2 − cos β2 ] = 0, and therefore, N3 (α, β1 , β2 , β3 , z1 , z2 , z3 , z4 ) =

1 cos β2 , sin β1 sin β2

as required. Equations (251), (256)–(258) are kinematic relations that define the coordinates α, β1 , β2 , β3 , z1 , z2 , z3 , z4 in the phase space of system (251)–(258) (the tangent bundle T∗ S4 {z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 }). 3.4.5. Transition in n: 5 → 6. When passing from n = 5 to n = 6, the first four variables are reindexed, ⎛ ⎞ ⎛ ⎞ z4 z5 ⎜ z3 ⎟ ⎜z4 ⎟ ⎜ ⎟ → ⎜ ⎟ , ⎝ z2 ⎠ ⎝z3 ⎠ z1 z2 and we introduce a new z1 . New terms that appear with the increase of n are underlined. 575

Proposition 3.4. For n = 6, the following system defines a geodesic flow on the tangent bundle T∗ S5 {z5 , z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 , β4 } of the five-dimensional sphere S5 {α, β1 , β2 , β3 , β4 }: α = −z5 ,

(264)

z5 = −(z42 + z32 + z22 + z12 )

cos α , sin α

(265)

z4 = z4 z5

cos α cos β1 cos α + (z32 + z22 + z12 ) , sin α sin α sin β1

(266)

z3 = z3 z5

cos α 1 cos β2 cos α cos α cos β1 − z3 z4 − (z2 + z12 ) , sin α sin α sin β1 sin α sin β1 sin β2

(267)

z2 = z2 z5

cos α cos α cos β1 cos α 1 cos β2 cos α 1 1 cos β3 + z2 z3 + z12 , − z2 z4 sin α sin α sin β1 sin α sin β1 sin β2 sin α sin β1 sin β2 sin β3

(268)

z1 = z1 z5

cos α cos α cos β1 cos α 1 cos β2 cos α 1 1 cos β3 − z1 z4 + z1 z3 − z1 z2 , sin α sin α sin β1 sin α sin β1 sin β2 sin α sin β1 sin β2 sin β3

(269)


β1 = z4

β3 = z2

(270) (271)

cos α , sin α sin β1 sin β2

β4 = −z1

(272)

cos α , sin α sin β1 sin β2 sin β3

(273)

and, in view of Remark 3.5, there exist the first integrals z12 + z22 + z32 + z42 + z52 = C1 = const, z12 + z22 + z32 + z42 sin α = C2 = const, z12 + z22 + z32 sin α sin β1 = C3 = const, z12 + z22 sin α sin β1 sin β2 = C4 = const.

(277)

z1 sin α sin β1 sin β2 sin β3 = C5 = const.

(278)

(274) (275) (276)

Indeed, using (274)–(277) and arguing as in the proof of Propositions 3.1–3.3, we find the underlined coefficients in equations (265)–(267) and conclude that equations (268), (269) can be written in the form cos α cos α cos β1 cos α 1 cos β2 − z2 z4 + z2 z3 sin α sin α sin β1 sin α sin β1 sin β2 2 cos α N4 (α, β1 , β2 , β3 , β4 , z1 , z2 , z3 , z4 , z5 ), + z1 sin α

z2 = z2 z5

z1

576

cos α cos α cos β1 cos α 1 cos β2 − z1 z4 = z1 z5 + z1 z3 sin α sin α sin β1 sin α sin β1 sin β2 cos α N4 (α, β1 , β2 , β3 , β4 , z1 , z2 , z3 , z4 , z5 ). − z1 z2 sin α

(279)

Due to (278), (264)–(273), we have z1 sin α sin β1 sin β2 sin β3 + z1 α cos α sin β1 sin β2 sin β3 + z1 β1 sin α cos β1 sin β2 sin β3 + z1 β2 sin α sin β1 cos β2 sin β3 + z1 β3 sin α sin β1 sin β2 cos β3 = z1 z2 cos α[cos β3 − N4 (α, β1 , β2 , β3 , β4 , z1 , z2 , z3 , z4 , z5 ) sin β1 sin β2 sin β3 ] = 0, and therefore, N4 (α, β1 , β2 , β3 , β4 , z1 , z2 , z3 , z4 , z5 ) =

1 1 cos β3 , sin β1 sin β2 sin β3

as required. Equations (264), (270)–(273) are kinematic relations that define the coordinates α, β1 , β2 , β3 , β4 , z1 , z2 , z3 , z4 , z5 in the phase space of system (264)–(273) (the tangent bundle T∗ S5 {z5 , z4 , z3 , z2 , z1 ; α, β1 , β2 , β3 , β4 }). 3.4.6. Transition in n: n → n + 1. The inductive transition from n to n + 1 involves the transformation ⎞ ⎞ ⎛ ⎛ zn zn−1 ⎜zn−1 ⎟ ⎜zn−2 ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ . . . ⎠ → ⎝ . . . ⎠ , z1 z2 the introduction of a new z1 , and underlining new terms appearing with the increase of n. Proposition 3.5. For n > 2, the following system defines a geodesic flow on the tangent bundle T∗ Sn {zn , zn−1 , . . . , z1 ; α, β1 , . . . , βn−1 } of the n-dimensional sphere Sn {α, β1 , . . . , βn−1 }: α = −zn ,

(280)

2 zn = −(zn−1 + · · · + z22 + z12 )

cos α , sin α

(281)

zn−1 = zn−1 zn

cos α cos β1 cos α 2 + · · · + z22 + z12 ) , + (zn−2 sin α sin α sin β1

(282)

= zn−2 zn zn−2

cos α 1 cos β2 cos α cos α cos β1 2 − zn−2 zn−1 − (zn−3 + · · · + z22 + z12 ) , sin α sin α sin β1 sin α sin β1 sin β2

(283)

... cos α cos α cos β1 − z2 zn−1 + z2 zn−2 sin α sin α sin β1 1 cos α 1 + · · · + (−1)n+1 z2 z3 ... sin α sin β1 sin βn−4

z2 = z2 zn

cos α 1 cos β2 sin α sin β1 sin β2 1 cos βn−3 cos α 1 cos βn−2 + (−1)n+1 z12 ... , sin βn−3 sin α sin β1 sin βn−3 sin βn−2 (284)

cos α cos α cos β1 cos α 1 cos β2 − z1 zn−1 + z1 zn−2 sin α sin α sin β1 sin α sin β1 sin β2 1 1 cos α 1 cos βn−3 cos α 1 cos βn−2 + · · · + (−1)n+1 z1 z3 ... + (−1)n z1 z2 ... , sin α sin β1 sin βn−4 sin βn−3 sin α sin β1 sin βn−3 sin βn−2 (285)

z1 = z1 zn

cos α , sin α cos α , β2 = −zn−2 sin α sin β1 ... β1 = zn−1

(286) (287)

577

βn−2 = (−1)n+1 z2 βn−1 = (−1)n z1


(288)


(289)

and, in view of Remark 3.5, there exist the first integrals z12 + · · · + zn2 = C1 = const, 2 z12 + · · · + zn−1 sin α = C2 = const, 2 z12 + · · · + zn−2 sin α sin β1 = C3 = const,

(290) (291) (292)

... z12 + z22 sin α sin β1 . . . sin βn−3 = Cn−1 = const.

z1 sin α sin β1 . . . sin βn−2 = Cn = const.

(293) (294)

Indeed, using (290)–(293) and arguing as in the proof of Propositions 3.1–3.4, we find the underlined coefficients in all equations up to (284) and (285), and conclude that equations (284), (285) can be written in the form cos α cos α cos β1 cos α 1 cos β2 − z2 zn−1 + z2 zn−2 z2 = z2 zn sin α sin α sin β1 sin α sin β1 sin β2 1 1 cos α cos βn−3 + · · · + (−1)n+1 z2 z3 ... sin α sin β1 sin βn−4 sin βn−3 n+1 2 cos α Nn−1 (α, β1 , . . . , βn−1 , z1 , . . . , zn ), + (−1) z1 sin α (295) cos β 1 cos α cos α cos α cos β 1 2 z1 = z1 zn − z1 zn−1 + z1 zn−2 sin α sin α sin β1 sin α sin β1 sin β2 1 1 cos α cos βn−3 + · · · + (−1)n+1 z1 z3 ... sin α sin β1 sin βn−4 sin βn−3 cos α n Nn−1 (α, β1 , . . . , βn−1 , z1 , . . . , zn ). + (−1) z1 z2 sin α Further, due to (294), (280)–(289), we have z1 sin α sin β1 . . . sin βn−2 + z1 α cos α sin β1 . . . sin βn−2

+ z1 β1 sin α cos β1 sin β2 . . . sin βn−2 + · · · + z1 βn−3 sin α sin β1 . . . sin βn−4 cos βn−3 sin βn−2 + z1 βn−2 sin α sin β1 . . . sin βn−3 cos βn−2

= z1 z2 cos α[cos βn−2 − Nn−1 (α, β1 , . . . , βn−1 , z1 , . . . , zn ) sin β1 . . . sin βn−2 ] = 0, and therefore, Nn−1 (α, β1 , . . . , βn−1 , z1 , . . . , zn ) =

1 cos βn−2 1 ... , sin β1 sin β2 sin βn−2

as required. Equations (280), (286)–(289) are kinematic relations defining the coordinates α, β1 , . . . , βn−1 , z1 , . . . , zn in the phase space of system (280)–(289) (the tangent bundle T∗ Sn {zn , . . . , z1 ; α, β1 , . . . , βn−1 }). 3.5. General Remarks on the Integrability for an Arbitrary n. As stated above, for the complete integrability of system (108)–(116) of order 2(n − 1), in general, it is necessary to know 2n − 3 independent first integrals. However, the systems under consideration have some symmetries that allow us to reduce the sufficient number of the first integrals to n. 578

3.5.1. System without a force field. Consider system (108)–(116) on 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 }. From this we are going to obtain a conservative system, assuming that the function (47) is identically equal to zero (in particular, b = 0 and there is no coefficient sin α cos α in equation (109)). The system under consideration takes the form α = −zn−1 ,

(296)

2 zn−1 = −(z12 + · · · + zn−2 )

cos α , sin α

(297)

= zn−2 zn−1 zn−2

cos α cos β1 cos α 2 + (z12 + · · · + zn−3 ) , sin α sin α sin β1

(298)

zn−3 = zn−3 zn−1


(299)

... z1

cos α = z1 sin α

n−2 (−1)s+1 zn−s s=1



= (−1)n z2 βn−3

(301) (302)


= (−1)n+1 z1 βn−2

(300)


(303) (304)

System (296)–(304) describes the motion of the rigid body in the absence of external forces. Theorem 3.10. System (296)–(304) admits n independent analytic first integrals of the form 2 = C1 = const, Φ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z12 + · · · + zn−1 2 sin α = C2 = const, Φ2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z12 + · · · + zn−2 Φ3 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 z12 + · · · + zn−3 sin α sin β1 = C3 = const,

... Φn−2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =


(305) (306) (307)

(308)

Φn−1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z1 sin α sin β1 . . . sin βn−3 = Cn−1 = const,

(309)

Φn (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(310)

The first n − 1 integrals (305)–(309) show that due to the absence of external forces four (nonzero, in general) components of the angular velocity tensor of a five-dimensional body are preserved: ωr1 ≡ ωr01 = const, . . . , ωrn−1 ≡ ωr0n−1 = const.

(311) 579

In particular, the existence of the first integral (305) is due to the relation 2 = ωr21 + · · · + ωr2n−1 ≡ C12 = const. z12 + · · · + zn−1

(312)

The last integral (310) has kinematic sense and “attaches” the equation for βn−2 . This integral can be found from the quadrature z1 1 dβn−2 =− . (313) dβn−3 z2 sin βn−3 Using the level constants of the first integrals (308), (309), we can write 2 Cn−2 z1 =± sin2 βn−3 − 1, (314) 2 z2 Cn−1 and therefore, the quadrature (313) takes the form du βn−2 = ± 2 Cn−2 (1 − u2 ) − 1 − C2 n−1

2 Cn−2 u2 2 Cn−1

,

u = cos βn−3 .

(315)

,

Cn = const,

(316)

Hence, we get βn−2 + Cn = ± arctan

cos βn−3 2 Cn−2 2 Cn−1

2

sin βn−3 − 1

which allows us to obtain the first integral (310). Transforming (316), we obtain the following invariant relation: 2 Cn−1 . (317) tan2 (βn−2 + Cn ) = 2 2 ) tan2 β 2 (Cn−2 − Cn−1 n−3 − Cn−1 Now let us restate Theorem 3.10. Theorem 3.11. System (296)–(304) admits n independent first integrals of the form: Ψ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 z 2 + · · · + zn−1 Φ21 = C1 = const, = 1 Φ2 2 2 z1 + · · · + zn−2 sin α

Ψ2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = C2 = const, z12 + z22 Φn−2 = = C3 = const, Ψ3 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Φn−1 z1 sin βn−3 ... 2 z12 + · · · + zn−3 Φ3 Ψn−2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = = = Cn−2 = const, Φ4 2 2 z + ··· + z sin β 1

Ψn−1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

n−4

2 z12 + · · · + zn−2

(319) (320)

(321)

2

Φ2 = = Cn−1 = const, Φ3 2 2 z1 + · · · + zn−3 sin β1

Ψn (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(318)

(322)

(323)

The last integral (323) also has kinematic sense and “attaches” the equations for βn−2 , while the functions Ψ2 , Ψn can be chosen equal to Φ2 , Φn , respectively. Theorem 3.11 (in contrast to Theorem 3.10) does not specify the smoothness of the first integrals. Thus, the points at which the denominators (or numerators and denominators simultaneously) of the first 580

integrals (318)–(323) vanish are singular points of these integrals regarded as functions, which can even be discontinuous, in general. Due to Theorem 3.11, the transformed set of the first integrals (318)–(323) of system (296)–(304) (without external forces) is still a set of its first integrals. For the 8th-order system (296)–(304) to be completely integrable, it is necessary to know its 2n − 3 independent first integrals. However, after the transformation of the variables ⎞ ⎞ ⎛ wn−1 zn−1 ⎜wn−2 ⎟ ⎜zn−2 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ . . . ⎟ → ⎜ . . . ⎟ , wn−1 = zn−1 , wn−2 = z 2 + · · · + z 2 , 1 n−2 ⎟ ⎟ ⎜ ⎜ ⎝ w2 ⎠ ⎝ z2 ⎠ z1 w1 z2 z3 zn−3 zn−2 wn−3 = , wn−4 = 2 , . . . , w2 = , w1 = , (324) 2 z1 2 2 2 2 z1 + z2 z1 + · · · + zn−4 z1 + · · · + zn−3 ⎛

system (296)–(304) splits into the equations α = −wn−1 ,

(325)

cos α , sin α cos α = wn−2 wn−1 , sin α

2 = −wn−2 wn−1

(326)

wn−2

(327)

1 + ws2 cos βs , ws sin βs βs = ds (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ), s = 1, . . . , n − 3,

(328)

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

(329)

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

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 )

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

(330)


and zk = Zk (wn−1 , . . . , w1 ),

k = 1, . . . , n − 2,

(331)

are the functions defined by (324). We see that system (325)–(329) of order 3 + 2(n − 3) + 1 = 2(n − 1) splits into independent subsystems of lower orders: the third-order system (325)–(327) and the second-order systems (328) (of course, after changing the independent variable). Thus for the complete integrability of system (325)–(329) it suffices to find two independent first integrals of system (325)–(327), one integral for each system (328) (n − 3 integrals altogether), and an additional first integral that “attaches” equation (329) (i.e., n altogether).

581

Remark 3.6. Let us write out the first integrals (318)–(323) in the variables w1 , . . . , wn−1 defined by (324). We get Θ1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) =

2 2 + wn−1 wn−2 = C1 = const, wn−2 sin α

Θ2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = wn−2 sin α = C2 = const, 1 + ws2 Θs+2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = = Cs+2 = const, sin βs

(332) (333) s = 1, . . . , n − 3,

Θn (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(334) (335)

Thus, two independent first integrals (332), (333) are sufficient for the integration of system (325)–(327); the first integrals (334) (n − 3 in number) are sufficient for the integration of the independent first-order equations 1 + ws2 cos βs dws = , s = 1, . . . , n − 3, (336) dβs ws sin βs which, after changing the independent variable, are equivalent to systems (328), respectively; and finally, the first integral (335) is sufficient for “attaching” equation (329). Thus, we have proved the following result. Theorem 3.12. System (296)–(304) of order 2(n−1) has sufficiently many (n) independent first integrals. 3.5.2. System with conservative forces. Consider system (108)–(116) with the condition b = 0. We thus obtain a conservative system. Indeed, the presence of forces is characterized by the coefficient sin α cos α in equation (109) (in contrast to system (296)–(304)). The system under consideration takes the form α = −zn−1 ,

(337)

2 zn−1 = sin α cos α − (z12 + · · · + zn−2 )

cos α , sin α

(338)

zn−2 = zn−2 zn−1

cos α cos α cos β1 2 ) , + (z12 + · · · + zn−3 sin α sin α sin β1

(339)

= zn−3 zn−1 zn−3


(340)

... z1

cos α = z1 sin α

n−2 (−1)s+1 zn−s s=1



= (−1)n z2 βn−3


(341) (342) (343)

(344)

cos α . (345) sin α sin β1 . . . sin βn−3 Thus, system (337)–(345) describes the motion of a rigid body in a conservative field of external forces. = (−1)n+1 z1 βn−2

582

Theorem 3.13. System (337)–(345) admits n independent first integrals of the form 2 + sin2 α = C1 = const, Φ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z12 + · · · + zn−1 2 sin α = C2 = const, Φ2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z12 + · · · + zn−2

Φ3 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 z12 + · · · + zn−3 sin α sin β1 = C3 = const,

... Φn−2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =


(346) (347) (348)

(349)

Φn−1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = z1 sin α sin β1 . . . sin βn−3 = Cn−1 = const,

(350)

Φn (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(351)

The first integral (346) is the integral of total energy. The last integral (351) has kinematic sense, it “attaches” the equation for βn−2 and has been found above. Let us restate Theorem 3.13. Theorem 3.14. System (337)–(345) admits n independent first integrals of the form Ψ1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

2 z12 + · · · + zn−1 + sin2 α Φ1 = C1 = const, = Φ2 2 2 z1 + · · · + zn−2 sin α

Ψ2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = C2 = const, z12 + z22 Φn−2 = = C3 = const, Ψ3 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Φn−1 z1 sin βn−3 ... 2 z12 + · · · + zn−3 Φ3 Ψn−2 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = = = Cn−2 = const, Φ4 2 2 z + ··· + z sin β 1

Ψn−1 (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) =

n−4

2 z12 + · · · + zn−2

(353) (354)

(355)

2

Φ2 = = Cn−1 = const, Φ3 2 2 z1 + · · · + zn−3 sin β1

Ψn (zn−1 , . . . , z1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(352)

(356)

(357)

The functions Ψ2 , Ψn can be chosen equal to Φ2 , Φn , respectively. Theorem 3.14 (in contrast to Theorem 3.13) does not characterize smoothness of the first integrals. Thus, the points at which the denominators (or numerators and denominators simultaneously) of the first integrals (352)–(357) vanish are points of singularity of these integrals regarded as functions, which can even be discontinuous at these points. Due to Theorem 3.14, the transformed set of the first integrals (352)–(357) of system (337)–(345) (with a conservative field of forces) is still a set of its first integrals. For the complete integration of system (337)–(345) of order 2(n − 1), in general, it is necessary to know 2n − 3 independent first integrals. However, after the transformation of the variables (324), system 583

(337)–(345) splits into the equations α = −wn−1 ,

(358)

2 wn−1 = sin α cos α − wn−2 wn−2 = wn−2 wn−1

cos α , sin α

(359)

cos α , sin α

(360)


(361)

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

(362)

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

and conditions (330) are satisfied. We see that system (358)–(362) of order 2(n − 1) splits into independent subsystems of lower orders: system (358)–(360) of order 3, and systems (361) (after changing the independent variable) of order 2. Thus, for the complete integrability of system (358)–(362), it suffices to find two independent first integrals of system (358)–(360), one integral for each system (361) (n−3 altogether), and an additional first integral that “attaches” equation (362) (altogether n). Remark 3.7. Let us write out the first integrals (352)–(357) in the variables w1 , . . . , wn−1 defined by (324). We get Θ1 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) =

2 2 wn−2 + wn−1 + sin2 α = C1 = const, wn−2 sin α

Θ2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = wn−2 sin α = C2 = const, 1 + ws2 = Cs+2 = const, Θs+2 (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = sin βs Θn (wn−1 , . . . , w1 ; α, β1 , . . . , βn−2 ) = Cn = const.

(363) (364)

s = 1, . . . , n − 3,

(365) (366)

Thus, two independent first integrals (363), (364) are sufficient for the integration of system (358)–(360), the first integrals (365) (n − 3 in number) are sufficient for the integration of the independent first-order equations 1 + ws2 cos βs dws = , s = 1, 2, (367) dβs ws sin βs which, after changing the independent variable, are equivalent to systems (361), respectively; and finally, the first integral (366) is sufficient for “attaching” equation (362). Thus, we have established the following result. Theorem 3.15. System (337)–(345) of order 2(n−1) has sufficiently many (n) independent first integrals. 3.6. Complete List of Invariant Relations for an Arbitrary n. Let us turn to the integration of the 2(n − 1)th-order system (108)–(116) without simplifications, with all coefficients. For the complete integration of the 2(n − 1)th-order system (108)–(116), in general, it is necessary to know 2n − 3 independent first integrals. However, after the transformation of the variables (324), system (108)–(116) splits as follows: α = −wn−1 + b sin α, 2 wn−1 = sin α cos α − wn−2

584


(369)

wn−2 = wn−2 wn−1

cos α , sin α

(370)


(371)

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

(372)

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

and conditions (330) are satisfied. We see that system (368)–(372) of order 2(n − 1) splits into independent subsystems of lower orders: system (368)–(370) of order 3 and systems (371) (after changing the independent variable) of order 2. Therefore, for the complete integration of system (368)–(372) it suffices to find two independent first integrals of system (368)–(370), one integral for each system (371) (altogether n − 3), and an additional first integral “attaching” equation (372) (i.e., n altogether). First, let us associate to the third-order system (368)–(370) the nonautonomous second-order system 2 sin α cos α − wn−2 cos α/ sin α dwn−1 = , dα −wn−1 + b sin α

(373)

wn−2 wn−1 cos α/ sin α dwn−2 = . dα −wn−1 + b sin α Changing the variable, τ = sin α, we rewrite system (373) in algebraic form, 2 /τ τ − wn−1 dwn−1 = , dτ −wn−1 + bτ

(374)

wn−2 wn−1 /τ dwn−2 = . dτ −wn−1 + bτ Introducing the homogeneous variables wn−2 = u1 τ,

wn−1 = u2 τ,

(375)

we reduce system (374) to τ

1 − u21 du2 + u2 = , dτ −u2 + b

(376)

u1 u2 du1 + u1 = , τ dτ −u2 + b which is equivalent to τ

1 − u21 + u22 − bu2 du2 = , dτ −u2 + b

2u1 u2 − bu1 du1 = . τ dτ −u2 + b Let us associate to the second-order system (377) the nonautonomous first-order equation du2 1 − u21 + u22 − bu2 = , du1 2u1 u2 − bu1 which can be easily reduced to the total differential 2 u2 + u21 − bu2 + 1 = 0. d u1

(377)

(378)

(379) 585

Thus, equation (378) has the following first integral: u22 + u21 − bu2 + 1 = C1 = const, u1

(380)

which, in the former variables, reads Θ1 (wn−1 , wn−2 ; α) =

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

(381)

Remark 3.8. For b = 0, the first integral (381) of system (368)–(370) coincides with the first integral (363) of system (358)–(360), but for b = 0, neither the numerator of the expression (381) nor its denominator are first integrals of system (368)–(370) separately (although, for b = 0, both the numerator and the denominator of (381) are first integrals of system (358)–(360)). Next, we find, in explicit form, the additional first integral for the third-order system (368)–(370). We start by transforming the invariant relation (380) for u1 = 0 as follows: b 2 C1 2 b2 + C12 u2 − + u1 − = − 1. (382) 2 2 4 We see that the parameters of this invariant relation must satisfy the condition b2 + C12 − 4 ≥ 0,

(383)

and the phase space of system (368)–(370) foliates into the family of surfaces defined by (382) in the coordinates u1 , u2 . Thus, in view of (380), the first equation of system (377) takes the form τ

2(1 − bu2 + u22 ) − C1 U1 (C1 , u2 ) du2 = , dτ −u2 + b

(384)

where

1 C1 ± C12 − 4(u22 − bu2 + 1) , (385) U1 (C1 , u2 ) = 2 and the integration constant C1 is chosen from the condition (383). Therefore, the quadrature that would allow us to find the additional first integral of system (368)–(370) takes the form (b − u2 ) du2 dτ = . (386) 2 τ 2 (1 − bu2 + u2 ) − C1 {C1 ± C12 − 4 (u22 − bu2 + 1)}/2 Obviously, the left-hand side (to within an additive constant) is equal to ln | sin α|.

(387)

For u2 −

b = r1 , 2

b21 = b2 + C12 − 4,

(388)

the right-hand side of (386) becomes

b d(b21 − 4r12 ) 1

b21 − 4r12 dr1 1

−b = − ln

± 1 ± I1 , (389) − 2 2 2 2 2 2 2 2

2 4 2 C 1 (b1 − 4r1 ) ± C1 b1 − 4r1 (b1 − 4r1 ) ± C1 b1 − 4r1 where

I1 =

dr3 , b21 − r32 (r3 ± C1 )

r3 =

b21 − 4r12 .

To calculate the integral (390), we consider three possible cases. 586

(390)

(I) b > 2.

√ 2

b − 4 + b21 − r32 1 C1

√ √ ln I1 = − ± r3 ± C1 2 b2 − 4

b2 − 4

√ 2

b − 4 − b21 − r32 C1

1

∓√ + √ ln + const. r3 ± C1 b2 − 4

2 b2 − 4

(II) b < 2. 1 ±C1 r3 + b21 + const. arcsin I1 = √ b1 (r3 ± C1 ) 4 − b2 (III) b = 2. b21 − r32 + const. I1 = ∓ C1 (r3 ± C1 ) Going back to the variable b r3 − , r1 = sin α 2 we finally obtain the following expressions for I1 . (I) b > 2.

√ 2

b − 4 ± 2r1 1 C1

I1 = − √ ±√ ln 2 b2 − 4 b21 − 4r12 ± C1 b2 − 4

√ 2

b − 4 ∓ 2r1

C 1 1

+ const. √ ∓ ln

2 + √ b2 − 4

2 b2 − 4 b1 − 4r12 ± C1 (II) b < 2.

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

(391)

(392)

(393)

(394)

(395)

(396)

(III) b = 2.

2r1 + const. (397) 2 C1 ( b1 − 4r12 ± C1 ) Thus, we have found the additional first integral for the third-order system (368)–(370) and obtained a complete set of the first integrals, which are transcendental functions of their phase variables. I1 = ∓

Remark 3.9. In the expression of the first integral just found, the constant C1 should be replaced by the left-hand side of the first integral (380). Then the additional first integral takes the form wn−1 wn−2 (398) , = C2 = const. Θ2 (wn−1 , wn−2 ; α) = ln | sin α| + G2 sin α, sin α sin α Thus, we have found two first integrals (381), (398) of the independent third-order system (368)–(370). It remains to obtain one first integral for each system (371) (altogether n − 3) and an additional first integral “attaching” equation (372). The desired first integrals coincide with the first integrals (365), (366), namely: 1 + ws2 = Cs+2 = const, s = 1, . . . , n − 3, (399) Θs+2 (ws ; βs ) = sin βs (400) Θn (wn−3 , . . . , w1 ; α, β1 , . . . , βn−2 ) = Cn = const. where Cn−2 , Cn−1 in the left-hand side of (400) should be replaced by the integrals (399) for s = n−4, n−3, respectively. Theorem 3.16. System (368)–(372) of order 2(n−1) has sufficiently many (n) independent first integrals (381), (398)–(400). 587

Thus, in the case under consideration, the system of dynamic equations (20)–(24), (27)–(36) with the condition (96) has n2 − n + 4 (n − 1)(n − 2) +n= , n > 2, 1+ 2 2 invariant relations: the analytic nonintegrable constraint (44); the cyclic first integrals (40), (41); the first integral (381), and also the first integral defined by (391)–(398) and having the form of a transcendental (in the sense of complex analysis) function of its phase variables, which can be expressed as a finite combination of elementary functions; finally, the transcendental first integrals (399), (400). Theorem 3.17. System (20)–(24), (27)–(36) with the conditions (44), (96), (40), (41) admits (n2 − n + 4)/2, n > 2, invariant relations (a complete set), of which n are transcendental functions in the sense of comp[lex analysis. All these relations can be written in terms of finite combinations of elementary functions. This work has been supported by the Russian Foundation for Basic Research (Grant 15-01-00848-a). REFERENCES 1. A. A. Andronov and L. S. Pontryagin, “Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937). 2. V. I. Arnold, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1989). 3. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow (1985). 4. O. I. Bogoyavlenskii, “Some integrable cases of the Euler equations,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986). 5. A. D. Bryuno, The Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979). 6. N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1970). 7. 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). 8. D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations of a rigid body with a fixed point in Rn ,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002). 9. D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of a generalized gyroscope in Rn ,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 37–41 (2003). 10. V. V. Golubev, Lectures on the Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1950). 11. B. A. Dubrovin, B. A. Krichever, and S. P. Novikov, “Integrable Systems. I,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat., 4, 179–284 (1985). 12. V. V. Kozlov, Methods of Qualitative Analysis in Solid Dynamics [in Russian], Izd. Mosk. Univ., Moscow (1980). 13. V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983). 14. Yu. I. Manin, “Algebraic aspects of differential equations,” J. Sov. Math., 11, 1–128 (1979). 15. Z. Nitecky, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975). 16. V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian], Nauka, Moscow (1967). 17. A. Poincaré, On Curves Defined by Differential Equations, OGIZ, Moscow (1947). 18. A. Poincaré, “New methods in celestial mechanics,” in: Selected Works, Vols. 1, 2, Nauka, Moscow (1971, 1972). 19. V. A. Samsonov and M. V. Shamolin, “On the problem of motion of a body in resistant media,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 51–54 (1989). 588

20. S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970). 21. V. A. Steklov, On the Motion of a Solid Body in Fluid [in Russian], Kharkiv (1893). 22. V. V. Trofimov and A. T. Fomenko, “Dynamical systems on linear representation orbits and complete integrability of some hydrodynamic systems,” Funkt. Anal. Pril., 17, No. 1, 31–39 (1983). 23. V. V. Trofimov and M. V. Shamolin, “Dissipative systems with nontrivial generalized Arnold–Maslov classes,” in: “Abstracts of lectures at the P. K. Rashevskii seminar on vector and tensor analysis,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 62 (2000). 24. Ph. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970). 25. S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow, 1976. 26. M. V. Shamolin, “Closed trajectories of different topological type in the problem of motion of a body in a resistant medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 52–56 (1992). 27. M. V. Shamolin, “A problem of motion of a body in a resistant medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 52–58 (1992). 28. M. V. Shamolin, “Classification of phase portraits in the problem of motion of a body in a resistant medium with linear damping moment,” Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993). 29. M. V. Shamolin, “An integrable case in spatial dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997). 30. M. V. Shamolin, “Spatial topographic Poincaré systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). 31. M. V. Shamolin, “On integrability in terms of transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). 32. M. V. Shamolin, “A family of portraits with limit cycles in the dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998). 33. M. V. Shamolin, “Some classes of particular solutions in the dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178–189 (1999). 34. M. V. Shamolin, “New Jacobi integrable cases in the dynamics of a rigid body interacting with media,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999). 35. M. V. Shamolin, “On roughness of dissipative systems and relative roughness and nonroughness of systems with variable dissipation,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999). 36. M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). 37. M. V. Shamolin, “Jacobi integrability in the problem of a four-dimensional rigid body in a resistant medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000). 38. M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in an incoming flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001). 39. M. V. Shamolin, “On the integrability of some classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). 40. M. V. Shamolin, “A completely integrable case in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005). 41. M. V. Shamolin, “Comparison of Jacobi integrable cases in plane and spatial dynamics of jet flow past a body,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005). 42. M. V. Shamolin, “An integrable case of dynamical equations on so(4) × Rn ,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). 43. M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in a medium flow with rotational derivatives of the torque produced by the medium taken into account,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 3, 187–192 (2007). 44. M. V. Shamolin, “A case of complete integrability in the dynamics on the tangent bundle of the two-dimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007). 45. M. V. Shamolin, Methods of Analysis of Dynamical Systems with Variable Dissipation in the Dynamics of Solids [in Russian], Examen, Moscow (2007). 589

46. M. V. Shamolin, “Dynamical systems with variable dissipation: approaches, methods, applications,” Fundam. Prikl. Mat., 14, No. 3, 3–237 (2008). 47. M. V. Shamolin, “Novel integrable cases in the dynamics of a body interacting with a medium taking into account dependence of the moment of the resistance force on the angular velocity,” Prikl. Mat. Mekh., 72, No. 2, 273–287 (2008). 48. M. V. Shamolin, “Integrability of some classes of dynamical systems in terms of elementary functions,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 43–49 (2008). 49. M. V. Shamolin, “A three-parameter family of phase portraits in the dynamics of rigid bodies interacting with media,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008). 50. M. V. Shamolin, “Classification of complete integrability cases in the dynamics of a four-dimensional rigid body in nonconservative field,” Sovr. Mat. Prilozh., 65, 132–142 (2009). 51. M. V. Shamolin, “New cases of complete integrability in the dynamics of a 4-dimensional dynamically symmetric rigid body in nonconservative field of forces,” Dokl. Ross. Akad. Nauk, 425, No. 3, 338–342 (2009). 52. M. V. Shamolin, “Integrability of some classes of nonconservative dynamical systems in terms of elementary functions,” Sovr. Mat. Pril., 62, 131–171 (2009). 53. M. V. Shamolin, “New integrable cases in spatial dynamics of rigid bodies,” Dokl. Ross. Akad. Nauk, 431, No. 3, 339–343 (2010). 54. M. V. Shamolin, “A case of complete integrability in the dynamics of a 4-dimensional rigid body in a nonconservative field,” Usp. Mat. Nauk, 65, No. 1, 189–190 (2010). 55. M. V. Shamolin, “New integrable cases in the dynamics of a 4-dimensional rigid body in a nonconservative field,” Dokl. Ross. Akad. Nauk, 437, No. 2, 190–193 (2011). 56. M. V. Shamolin, “A new case of complete integrability of dynamical equations on the tangent bundle of a three-dimensional sphere,” Vestn. Samarsk. Univ. Estestv. Ser., No. 5 (86), 187–189 (2011). 57. M. V. Shamolin, “A complete list of first integrals in the problem of motion of a 4-dimensional rigid body in nonconservative field with linear damping,” Dokl. Ross. Akad. Nauk, 440, No. 2, 187–190 (2011). 58. M. V. Shamolin, “Some questions of the qualitative theory in the dynamics of systems with variable dissipation,” Sovr. Mat. Pril., 78, 138–147 (2012). 59. M. V. Shamolin, “A new integrable case in the dynamics of a 4-dimensional rigid body in nonconservative field with linear damping,” Dokl. Ross. Akad. Nauk, 444, No. 5, 506–509 (2012). 60. M. V. Shamolin, “A new integrable case in spatial dynamics of a rigid body interacting with media with linear damping,” Dokl. Ross. Akad. Nauk, 442, No. 4, 479–481 (2012). 61. M. V. Shamolin, “A complete list of first integrals for the dynamical equations of motion of a rigid body in resistant media, with linear damping taken into account,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 44–47 (2012). 62. M. V. Shamolin, “A comparison of complete integrability cases in the dynamics of 2- and 4-dimensional rigid bodies in nonconservative field,” Sovr. Mat. Pril., 76, 84–99 (2012). M. V. Shamolin Moscow State University, Moscow, Russia E-mail: [email protected]

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INTEGRABLE SYSTEMS ON THE TANGENT

INTEGRABLE SYSTEMS ON THE TANGENT

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