ISSN 0001-4346, Mathematical Notes, 2014, Vol. 95, No. 3, pp. 308–315. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A. V. Borisov, I. S. Mamaev, A. V. Tsyganov, 2014, published in Matematicheskie Zametki, 2014, Vol. 95, No. 3, pp. 340–351.
On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics A. V. Borisov1* , I. S. Mamaev2** , and A. V. Tsyganov3*** 1 Udmurt State University, Izhevsk, Russia Machine Construction Institute, Russian Academy of Sciences, Moscow, Russia Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Ekaterinburg, Russia 2 Udmurt State University, Izhevsk, Russia Machine Construction Institute, Russian Academy of Sciences, Moscow, Russia Russian Academy of Sciences, Ekaterinburg, Russia 3 St. Petersburg State University, St. Petersburg, Russia
Received May 28, 2013
Abstract—Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all. DOI: 10.1134/S0001434614030031 Keywords: Poisson bracket, nonholonomic system, Poisson structure, dynamical system, conformally Hamiltonian representation, Casimir function, Routh sphere, rolling of a Chaplygin ball.
INTRODUCTION From the mathematical point of view, any dynamical system in the phase space M , dim M = 2n, with coordinates x1 , . . . , x2n , is determined by the equations of motion x˙ i = Xi ,
i = 1, . . . , 2n.
(0.1)
This system of differential equations specifies the vector field X=
2n
Xi
i=1
∂ , ∂xi
(0.2)
which is a linear operator on the space of smooth functions on the manifold M determining the change of any characteristic of the system F˙ = X(F ) =
2n i=1
Xi
∂F ∂xi
if the dynamical equations (0.1) hold. In order to study the properties of the vector field X, we can decompose it into smooth vector fields Xk , X = g1 X1 + · · · + gm Xm , *
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corresponding to some known subgroup Diff(M ). By Klein’s Erlangen program, the choice of such a subgroup involves the definition of a “geometry” or of a “geometric structure” on the manifold M . For example, in special relativity theory, this will be the Poincare´ group; in Hamiltonian dynamics, the group of simplectomorphisms; in quantum mechanics, the unitary group; as well as the group of contact transformations, etc. In all of these cases, the subgroup of the diffeomorphism group defined by geometry, characterizes the geometry itself. In classical mechanics, decompositions into collections of commuting vector fields, X = g1 X1 + · · · + gn Xn ,
(0.3)
are also considered; here the Xk are either symmetry fields or Hamiltonian vector fields Xk = P dHk constructed by using the functionally independent first integrals Hi , H˙ i = X(Hi ) = 0,
i = 1, . . . , n,
and using the Poisson bivector P with respect to which the integrals of motion are in involution {Hi , Hj } =
2n
Pkm
km
∂Hi ∂Hj = 0. ∂xk ∂xm
(0.4)
The decomposition (0.3) can be called a generalized conformally Hamiltonian representation. The existence of such a decomposition and, in turn, the existence of a globally defined Poisson structure is an additional qualitative characteristic describing the given vector field. For example, this Poisson structure allows us to identify the level surfaces of integrals of motion with Lagrangian submanifolds and obtain the relevant conclusions. If a connected component of the level of integrals of motion is diffeomorphic to the two-dimensional torus and, on this torus, we have a dynamical system with invariant measure, then, by Kolmogorov’s theorem [1], we can introduce coordinates ϕ1 , ϕ2 (mod 2π) in which this system takes the form ϕ˙ 1 =
c1 , Φ
ϕ˙ 2 =
c2 . Φ
Here Φ is a smooth positive function on the torus and c1,2 are constants. The change of time dt → Φ dτ allows us to linearize the equations of motion and, therefore, the motion occurs along the rectilinear windings of the torus, although, possibly, not uniformly. In the variables ϕ1,2 , the vector field is conformally Hamiltonian X = g1 P dH1 ,
g1 = Φ−1 .
(0.5)
A generalization of Kolmogorov’s theorem in a neighborhood of a nonsingular two-dimensional torus and obstacles on the way to the global conformal Hamiltonian property are discussed in [2]. In classical Hamiltonian dynamics, g1 = const, while the Poisson structure (0.4) is usually at most linear and is defined by a sum of semisimple Lie algebras for which the Casimir functions are polynomials [3]. For nonholonomic systems (considered further), gi = const, while the Poisson structure (0.4) is essentially nonlinear and degenerate and its Casimir functions can be expressed in terms of complicated transcendental functions or not appear at all (in the global sense, just as first integrals of equations of motion). The absence of global Casimir functions for Suslov’s nonholonomic problem was first noted in [4]. The systems considered below once again indicate that the hierarchy of the dynamical behavior of nonholonomic systems introduced in [5], [6] possesses finer levels, and particular nonholonomic systems themselves can induce meaningful nonlinear Poisson structures whose general theory is still, unfortunately, poorly developed. For example, in contrast to the existence of integrals of motion, there is still no theory dealing with the topological, analytic, or algebraic obstacles to the existence of a Casimir functions. MATHEMATICAL NOTES
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1. ROLLING OF A BODY OF REVOLUTION ON A PLANE Consider the rolling of a body of revolution on a plane without slipping, i.e., when the velocity of the contact point is zero v + ω × r = 0,
(1.1)
where ω is the angular velocity, v is the velocity of the mass center of the body, and r is the vector joining the mass center to the contact point with respect to the moving coordinate systems related to the principal axes of the ball. The notation (a, b) denotes the usual inner product and a × b is the vector product of three-dimensional vectors. After the exclusion of the Lagrange multiplier corresponding to the nonholonomic constraint, the equations of motion consist of the equation for the kinetic momentum vector M with respect to the contact point and the kinematic equation for the unit normal vector at the contact point M˙ = M × ω + mr˙ × (ω × r) + MF ,
γ˙ = γ × ω,
(1.2)
where MF is the moment of external forces acting on the body. In the moving coordinate system, the angular momentum is related to the angular velocity by the equation M = IQ ω,
where
IQ = I + mr 2 E − mr ⊗ r.
(1.3)
Here m is the mass of the body, I is the central inertia tensor, and E is the unit matrix. In these equations, we assume that the vector r is related to the normal vector γ by the equality γ=−
grad f , |grad f |
defining the Gauss map for the function f (r) = 0 defining the surface of the body. A detailed proof of these equations of motion is given in [5]–[7]. If the body rolls on a plane under the action of potential forces, for example, of the force of gravity, then, in the first of Eqs. (1.2), MF = γ ×
∂U . ∂γ
In this case, Eqs. (1.2) possess the integrals of motion H=
1 (M, ω) + U (γ), 2
(1.4)
C = (γ, γ) = 1.
Following [8], we assume that the potential U depends only on γ3 , i.e., depends only on the angle between the axis of rotation of the body and the vertical. In this case, if the surface of the body and the central ellipsoid of the inertia are coaxial rotation surfaces, i.e., I1 = I2 = I3 ,
r1 = f1 (γ3 )γ1 ,
r2 = f1 (γ3 )γ2 ,
r3 = f2 (γ3 ),
(1.5)
then the vector field X defined by the equations of motion (1.2), possesses the symmetry field XS = γ1
∂ ∂ ∂ ∂ − γ2 + M1 − M2 , ∂γ2 ∂γ1 ∂M2 ∂M1
(1.6)
two additional integrals of motion, and the invariant measure g(γ) = I1 I3 + m(r, Ir). μ = g−1 (γ) dγ dM, It is well known [8], [5] that additional integrals of motion exist and are at most linear functions of the momenta Mi and real analytic, but not algebraic, functions of the coordinate γ3 : (k)
(k)
Jk = v1 (γ3 )(γ1 M1 + γ2 M2 ) + v2 (γ3 )M3 ,
k = 1, 2.
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(k)
The functions v1,2 appearing in this definition satisfy the equations (k)
g2 (v2
(k)
+ v1 ) (k) (k) = mf1 f2 ((1 − γ32 )f1 + γ3 f2 )(v2 f1 − v1 f2 ) m (k) (k) (k) (k) (k) + I1 f1 (1 − γ32 )(v1 f2 − v2 f1 ) − γ3 v1 f2 + (v1 (γ32 − 1) + γ3 v2 )f1 , (k)
g2 v1 m
= mf12 ((1 − γ32 )f1 + γ3 f2 )(v2 f1 − v1 f2 ) + I3 (f12 v2 − (v1 f2 − v2 f1 )f2 ). (k)
(k)
(k)
(k)
(k)
(1.8)
Here and elsewhere, we do not explicitly indicate the dependence of functions on γ3 . As an example, consider the disk whose mass center is displaced along the dynamical symmetry axis f1 =
R 1 − γ32
,
f2 = a.
Here R is the radius of the disk and a is the displacement of the mass center. If a = 0, then the functions (k) v1,2 are given by the relations (k)
(k)
v2 = c(L(k) (b+ , γ3 ) − γ3 L(k) (b− , γ3 )),
v1 = L(k) (b− , γ3 ), where
b∓ =
g2 − 4mI3 R2 1/2
2gI1
1 ∓ , 2
c=−
1/2 2 g
I1
k = 1, 2,
− 4mI3 R2 + gI1 , 2mgR2
are expressed in terms of the Legendre functions L(1,2) (bk , γ3 ) of the first and second kind, which, in turn, are defined by the hypergeometric function in the standard way [5]. If a = 0, then Eqs. (1.8) have no algebraic solutions. 2. THE POISSON STRUCTURE It is easy to verify that, for a body of rotation, Eq. (0.4) has a particular solution of the form ⎛ ⎞ 0 Γα ⎠ k = 1, 2, , rank Pα(k) = 2, Pα(k) = ζk ⎝ −Γα Mα where
⎞ (k) (k) γ1 γ2 v2 γ12 v2 γ2 ⎟ ⎜ 2 γ12 + γ22 v (k) ⎟ ⎜ γ1 + γ22 v (k) 1 1 ⎟ ⎜ ⎟ ⎜ (k) (k) 2 ⎟, ⎜ Γα = ⎜ γ2 v2 γ1 γ2 v2 ⎟ ⎜− γ 2 + γ 2 (k) − γ 2 + γ 2 (k) γ1 ⎟ ⎟ ⎜ 1 2 v1 1 2 v1 ⎠ ⎝ 0 0 0 ⎛
and
ˆ ζk = exp
m
⎞ (k) γ1 M1 + γ2 M2 v2 −M2 ⎟ ⎜0 (k) ⎟ ⎜ γ12 + γ22 v1 ⎟ ⎜ ⎟, ⎜ Mα = ⎜ 0 M1 ⎟ ⎟ ⎜∗ ⎠ ⎝ ∗ ∗ 0 ⎛
(k)
(k)
v1 g2 I1 v1 f1 ((1 − γ32 )f1 − γ3 f1 ) + I3 v2 (f1 f2 − f12 ) (k)
+
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dγ3 .
(2.1)
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If the condition of compatibility with the particular solution Pα obtained earlier is added to Eqs. (0.4), then it is easy to find another particular solution that will depend only on the form of the body, namely, ⎛ ⎞ 0 Γβ ⎠, Pβ = ⎝ (2.2) rank Pβ = 2, −Γβ Mβ where
⎞ γ1 γ2 γ3 γ12 γ3 ⎜ γ 2 + γ 2 − γ 2 + γ 2 0⎟ ⎟ ⎜ 1 2 1 2 ⎟ ⎜ 2 ⎟ ⎜ Γβ = ⎜ γ2 γ3 − γ1 γ2 γ3 0⎟ , ⎜ γ2 + γ2 2 2 γ1 + γ2 ⎟ ⎟ ⎜ 1 2 ⎠ ⎝ γ1 0 −γ2
and
⎛
⎛
⎜0 −M3 + ⎜ ⎜ Mβ = ⎜ ⎜∗ ⎝ ∗
⎞ γ3 (γ1 M1 + γ2 M2 ) + mρ −mσγ 2⎟ γ12 + γ22 ⎟ ⎟ ⎟, 0 mσγ1 ⎟ ⎠ ∗ 0
f1 (I3 f2 (γ1 M1 + γ2 M2 ) − (γ12 + γ22 )I1 f1 M3 ) f2 f2 σ− 1− , ρ= 2 f1 g f1 σ=
(γ1 M1 + γ2 M2 )(m(r, γ)f13 + I3 (f12 + f1 f2 )) g2 M3 (m(r, γ)f12 f2 + (γ3 f1 − (γ12 + γ22 )f1 )I1 f1 ) + . g2
The linear combination of the obtained particular solutions yields the desired complete solution of Eqs. (0.4) P (k) = αPα(k) + βPβ ,
rank P (k) = 4,
(2.3)
k = 1, 2,
where α and β are arbitrary functions of γ2 /γ1 and γ3 , respectively. Thus, we have two implicitly defined Poisson structures along with additional linear integrals of motion (1.7), which are their Casimir functions. In this case, the original vector field can be decomposed into a Hamiltonian field and a symmetry field X = β −1 P (k) dH + ηk XS ,
u1 (γ1 M1 + γ2 M2 ) + u2 M3 , g2
ηk =
(2.4)
where the coefficients of the decomposition mf1 (1 − γ32 )(β −1 αζk (v1 f2 − v2 f1 ) + v1 (γ3 f1 − f2 )) (k)
u1 =
(k)
(k)
(1 − γ32 )v1 I3 (γ3 v1 + β −1 αζk v2 ) (k)
+
(k)
(k)
(1 − γ32 )v1
,
mf2 (β −1 αζk (v1 f2 − v2 f1 ) + v1 (γ3 f1 − f2 )) − I1 v1 (β −1 αζk + 1) (k)
u2 =
(k)
(k)
(k)
(k)
(k)
,
v1 (k)
depend on the pairs of functions v1,2 appearing in the definition of the integrals of motion Jk (1.7). We can separate the symmetry field in the decomposition (2.4) by passing to the variables that are integrals for the symmetry field XS . Following [5], for such variables we choose the coordinate γ3 on which the potential U and the moment K1 =
(M, r) , f1
K2 = gω3 ,
K3 =
κ(γ1 M2 − γ2 M1 ) 1 − γ32
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κ=
313
1 − γ33 . I1 + m(r, r)
The bivectors P (k) have one general projection on the four-dimensional subspace with the coordinates x = (γ3 , K1 , K2 , K3 ), which is invariant with respect to the symmetry fields XS : ⎛ ⎞ 0 0 0 κ ⎜ ⎟ ⎜ ⎟ f2 ⎜ ⎟ K2 ⎟ ⎜0 0 0 −κI3 g−1 1 − f1 ⎜ ⎟ P = ⎜ rank P = 2. (2.6) ⎟, ⎜ ⎟ ⎜0 0 0 −κmg−1 f1 (f1 − f )K1 ⎟ ⎜ ⎟ 2 ⎝ ⎠ ∗ ∗ ∗ 0 The linear integrals of motion Jk (1.7) will be the Casimir functions of this bivector. It was this Poisson structure that was obtained in [5] by using Chaplygin’s reducing multiplier. For β = 1, the projection of the original vector field X (1.2), (2.4) is the Hamiltonian vector field = P dH. X Let us write out the first three equations of motion: f2 −1 K2 K3 , γ˙ 3 = κK3 , K˙ 1 = −κg I3 1 − f1
K˙ 2 = −κmg−1 f1 (f1 − f2 )K1 K3 .
If we divide the second and third equations by the first, then we obtain the linear nonautonomous firstorder equations dK2 f2 dK1 −1 = −g I3 1 − = −mg−1 f1 (f1 − f2 )K1 . (2.7) K2 , dγ3 f1 dγ3 These equations, in slightly different notation, were obtained by Chaplygin [8]; see a detailed discussion in [5]. The general solution of these equations can be expressed as the linear superposition Ki = c1 Φ1 (γ3 ) + c2 Φ2 (γ3 ),
i = 1, 2,
(2.8)
where the Φ1,2 are the fundamental solutions of the system of equations (2.7) and the constants c1,2 are, indeed, the values of linear (in momenta) integrals of motion H2,3 , which are the Casimir functions for the degenerate Poisson bivector P. 3. THE ROUTH SPHERE Following [5], [8], [9], we consider the rolling of an unbalanced dynamically symmetric ball on a plane without slipping. By an unbalanced ball we mean a ball whose mass center does not coincide with the geometric center, while, for a symmetric ball, two moments of inertia coincide with each other. For the Routh sphere, the vector r is of the form r = (Rγ1 , Rγ2 , Rγ3 + a), where R is the radius of the ball and a is the distance from the geometric center to the mass center of the ball. In this case, the vector field X (1.2) possesses the integrals HJ = K1 = (r, M ),
HR = K2 = g−1 (γ)ω3 .
The first of them is a Jellet integral; this integral also occurs in the case of a sufficiently general law for friction at the contact point. The second linear (in velocities) integral was obtained by Routh in 1884 and also by Chaplygin (see [8]). MATHEMATICAL NOTES
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Let us write out explicitly the Poisson bivector PJ for which the Jellet integral is a Casimir function. This bivector is parametrized by two functions α(γ1 /γ2 ) and β(γ3 ): ⎛ ⎞ ⎛ ⎞ 0 Γ 0 Γ α⎠ β⎠ +β⎝ . (3.1) PJ = αg ⎝ −Γα Mα −Γβ Mβ The matrices Γα,β are of the form ⎞ ⎛ γ1 γ2 (Rγ3 + a) γ22 (Rγ3 + a) −γ2 ⎟ ⎜ R(γ 2 + γ 2 ) R(γ12 + γ22 ) ⎟ ⎜ 1 2 ⎟ ⎜ ⎟ ⎜ γ12 (Rγ3 + a) γ γ (Rγ + a) 1 2 3 , Γα = ⎜ − γ1 ⎟ ⎟ ⎜ R(γ 2 + γ 2 ) − R(γ 2 + γ 2 ) ⎟ ⎜ 1 2 1 2 ⎠ ⎝ 0 0 0
⎞ γ1 γ2 γ3 γ12 γ3 ⎜− − γ 2 + γ 2 γ 2 + γ 2 0⎟ ⎟ ⎜ 1 2 1 2 ⎟ ⎜ 2 ⎟ ⎜ γ γ γ γ γ 3 1 2 3 2 Γβ = ⎜ − ⎟, 0 ⎜ γ12 + γ22 γ12 + γ22 ⎟ ⎟ ⎜ ⎠ ⎝ −γ1 0 γ2 ⎛
while the skew-symmetric matrices Mα,β are ⎞ ⎛ (γ1 M1 + γ2 M2 )(Rγ3 + a) −M2 ⎟ ⎜0 R(γ12 + γ22 ) ⎟ ⎜ ⎟ ⎜ Mα = ⎜ ⎟, 0 M1 ⎟ ⎜∗ ⎠ ⎝ ∗ ∗ 0 ⎞ ⎛ γ3 (γ1 M1 + γ2 M2 ) mσR(Rγ3 + a) mσγ2 R2 − ⎟ ⎜0 M3 − g2 g2 γ12 + γ22 ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ mσγ R 1 Mβ = ⎜∗ ⎟, 0 − ⎟ ⎜ g2 ⎟ ⎜ ⎠ ⎝ ∗ ∗ 0 where σ = mR(m(r, γ)C2 + I3 (γ1 M1 + γ2 M2 ) + I1 M3 γ3 ). By [9], the momenta are transformed as follows: 1 γ1 γ3 (R(γ1 M1 + γ2 M2 ) − bI1−1 (I1 + m(Rγ3 + a)2 ) L1 = 2 αg(Rγ3 + a) γ1 + γ22
γ2 (γ2 M1 − γ1 M2 ) + cγ1 , + β 1 γ2 γ3 (R(γ1 M1 + γ2 M2 ) − bI1−1 (I1 + m(Rγ3 + a)2 ) L2 = 2 αg(Rγ3 + a) γ1 + γ22
γ1 (γ2 M1 − γ1 M2 ) + cγ2 , − β M3 bm(Rγ3 + a) + , L3 = αg αgI1 where b = (M, r) and c = (L, γ), takes the Poisson bracket { · , · }J to the canonical Lie–Poisson bracket on the algebra e∗ (3), {Li , Lj }0 = εijk Lk ,
{Li , γj }0 = εijk γk ,
{γi , γj }0 = 0,
(3.2)
where εijk is a totally antisymmetric tensor. MATHEMATICAL NOTES
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If c = (γ, L) = 0, then the original integrals of motion are of the form
2 α2 g − I1 I3 2 g2 (Rγ3 + a)2 (L1 γ2 − L2 γ1 )2 H= 2 L3 + mR2 γ1 + γ22 R2 γ32 (I1 + mr 2 ) −
2bαg(Rγ3 + a)L3 b2 (I1 + m(Rγ3 + a)2 ) + , I1 R2 (γ12 + γ22 ) I12 R2 (γ12 + γ22 )
(3.3)
HR = αI1 L3 , so that the corresponding Hamiltonian equations are integrated absolutely in exactly the same way as for the Lagrange top [3]. ACKNOWLEDGMENTS This work was supported by the Grant of the Government of the Russian Federation for sponsoring scientific research (grant no. 11. G34.31.0039), by the Federal Target Program “Scientific and Pedagogical Personnel of Innovative Russia” (contract no. 2009-1.5-503-004-019), and by the program “Leading Scientific Schools” (grant no. NSh-2519.2012.1 “Dynamical Systems of Classical Mechanics and Control Problems”). REFERENCES 1. A. N. Kolmogorov, “On dynamical systems with an integral invariant on the torus,” Dokl. Akad. Nauk SSSR 93 (5), 763–766 (1953). 2. A. V. Bolsinov, A. V. Borisov, and I. S. Mamaev, “Hamiltonization non-holonomic systems in the neighborhood of invariant manifolds,” Regul. Chaotic Dyn. 16 (5), 443–464 (2011). 3. A. V. Borisov and I. S. Mamaev, Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed. (IKI, Moscow–Izhevsk, 2005) [in Russian]. 4. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “The Hamiltonian property and integrability of the Suslov problem,” Nonlinear Dynamics 6 (1), 127–142 (2010). 5. A. V. Borisov and I. S. Mamaev, “Rolling a rigid body on a plane and sphere. Hierarchy of dynamics,” Regul. Chaotic Dyn. 7 (2), 177–200 (2002). 6. A. V. Borisov, I. S. Mamaev, and A. A. Kilin, “Rolling a ball on a surface. New integrals and hierarchy of dynamics,” Regul. Chaotic Dyn. 7 (2), 201–219 (2002). 7. A. V. Borisov and I. S. Mamaev, “Conservation laws, hierarchy of dynamics and explicit integration nonholonomic systems,” Regul. Chaotic Dyn. 13 (5), 443–490 (2008). 8. S. A. Chaplygin, “On the motion of a heavy body of revolution on a horizontal plane,” in Collected Works (Moscow–Leningrad, 1948), Vol. 1 [in Russian]. 9. I. A. Bizyaev and A. V. Tsyganov, “On the Routh sphere,” Nonlinear Dynamics 8 (3), 569–583 (2012).
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