HAMILTONIZATION AND INTEGRABILITY OF THE CHAPLYGIN ...
arXiv:0902.4397v1 [math-ph] 25 Feb 2009
HAMILTONIZATION AND INTEGRABILITY OF THE CHAPLYGIN SPHERE IN Rn ˇ ´ BOZIDAR JOVANOVIC
Abstract. We prove that the n-dimensional Chaplygin–sphere problem for the zero value of the SO(n − 1)–momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.
1. Introduction Nonholomic systems are not Hamiltonian. Apparently, Chaplygin was one of the first who considered a time reparametrization in order to transform nonholonomic systems to the Hamiltonian form [9]. Also, after [8], one of the most famous solvable problems in nonholonomic mechanics, describing the rolling without slipping of a balanced ball over a horizontal surface, is refered as the Chaplygin sphere, see [1, 15]. It is interesting that the Hamiltonization of the system by the use of a time reparametrization is done just recently by Borisov and Mamaev [5, 6]. Fedorov and Kozlov constructed natural n-dimensional model of the Chaplygin– sphere problem and found an invariant measure [16]. Various aspects of the problem are studied in [23, 14, 19]. In [19], it is proved that the reduced equations of motion of the homogeneous ball are already Hamiltonian. However, the general problem of integrability and Hamiltonization is still unsolved. 1.1. Natural Nonholonomic Systems. Let Q be a n–dimensional Riemannian manifold with a nondegenerate metric κ(·, ·), V : Q → R be a smooth function and let D be a nonintegrable (n − k)–dimensional distribution of the tangent bundle T Q. A smooth path q(t) ∈ Q, t ∈ ∆ is called admissible (or allowed by constraints) if the velocity q(t) ˙ belongs to Dq(t) for all t ∈ ∆. Let q = (q1 , . . . , qn ) be some local coordinates on Q in which the constraints are written in the form n X αji q˙i = 0, j = 1, . . . , k, (1) (αjq , q) ˙ = i=1
where αj are independent 1-forms. The admissible path q(t) is a motion of the natural mechanical nonholonomic system (Q, κ, V, D) (or a nonholonomic geodesic for V ≡ 0) if it satisfies the Lagrange–d’Alambert equations k
(2)
d ∂L ∂L X λj αj (q)i , = + dt ∂ q˙i ∂qi j=1
MSC: 37J60, 37J35, 70H45 1
i = 1, . . . , n.
ˇ ´ BOZIDAR JOVANOVIC
2
Here the Lagrange multipliers λj are chosen such that the solutions q(t) satisfy constraints (1) and the Lagrangian is given by the difference of the kinetic and Pk P potential energy: L(q, q) ˙ = 12 ij κij q˙i q˙j − V (q). The expression j=1 λj αj (q)i represents the reaction forces of the constraints (1). P After the Legendre transformation pi = ∂L/q˙i = j κij q˙j one can also write the Lagrange-d’Alambert equations as a first-order system on the cotangent bundle k
∂H X λj αj (q)i , i = 1, . . . , n, + ∂qi j=1 P where the Hamiltonian is H(q, p) = 21 ij κij pi pj + V (q). As for the Hamiltonian systems, it is the first integral of the system. q˙i =
∂H , ∂pi
p˙ i =
1.2. Symmetries. Suppose that a Lie group K acts by isometries on (Q, κ) preserving the potential function V (the Lagrangian L is K- invariant) and let ξQ be the vector field on Q associated to the action of one-parameter subgroup exp(tξ), ξ ∈ k = TId K. The following version of the Noether theorem holds (see [1, 3]): if ξQ is a section of the distribution D then d ∂L d (3) , ξQ = (p, ξQ ) = 0. dt ∂ q˙ dt
On the other side, let ξQ be transversal to D, for all ξ ∈ k. In addition, suppose that Q has a principal bundle structure π : Q → Q/K and that D is the collection of horizontal spaces of a principal connection. Then (Q, κ, V, D) is called a KChaplygin system. The system (2) is K-invariant and reduces to the tangent bundle T (Q/K) = D/K (for the details see [20, 3, 7, 25]). In some cases the equations (2) have a rather strong property – an invariant measure (e.g, see [1, 4]). Within the class of K-Chaplygin systems, the existence of an invariant measure is closely related with their reduction to a Hamiltonian form after an appropriate time rescaling dτ = N dt (see [9, 24, 17, 7, 25, 11]). Nonholonomic systems on unimodular Lie groups with right-invariant constraints and left-invariant metrics, so called LR systems, always have an invariant measure [26]. Recently, a nontrivial example of a nonholonomic LR system on the group SO(n) (n-dimensional Veselova problem), which can be regarded also as a SO(n − 1)–Chaplygin system such that the reduced system on S n−1 = SO(n)/SO(n − 1) is Hamiltonian after the time rescaling, is given in [17].
1.3. Outline and Results of the Paper. In Section 2, we recall the equations of motion of the Chaplygin Sphere. The reduction of the system to the cotangent bundle of the sphere T ∗ S n−1 for a zero value of the SO(n − 1) momentum mapping is described in Section 3. The calculation of an invariant measure as well as the time reparametrization and the reduction of the system to the Hamiltonian form for a specific choice of an inertia operator of the ball is given in Section 4. It appears that the obtained Hamiltonian system is an integrable geodesic flow. Moreover, as in the 3-dimensional case [12],
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
3
the reduced system is closely related to the reduced nonholonomic Veselova problem (Section 5). 2. Chaplygin Sphere 2.1. Cinematics. Following [16, 14], consider the Chaplygin–sphere problem of rolling without slipping of an n-dimensional balanced ball (the mass center C coincides with the geometrical center) of radius ρ on an (n − 1)-dimensional hyperspace H in Rn . For the configuration space we take the direct product of Lie groups SO(n) and Rn , where g ∈ SO(n) is the rotation matrix of the sphere (mapping frame attached to the body to the space frame) and r ∈ Rn is the position vector of its center C (in the space frame). For a trajectory (g(t), r(t)) define angular velocities of the sphere in the moving and the fixed frame as well as the velocity in the fixed frame by ω = g −1 g, ˙ Ω = gg ˙ −1 , W = r. ˙ The Lagrangian of the system is then given by 1 1 (4) L = hIω, ωi + m(W, W). 2 2 Here I : so(n) → so(n) and m are the inertia tensor and mass of the ball, h·, ·i is given by 1 (5) hX, Y i = − tr(XY ), 2 and (·, ·) is the Euclidean scalar product. Let Γ ∈ Rn be a vertical unit vector (considered in the fixed frame) orthogonal to the hyperplane H and directed from H to the center C. The condition for the sphere to role without slipping leads that the velocity of the contact point is equal to zero: W − ρΩΓ = 0 .
(6) The distribution (7)
D = {(g, r, ω, W) | W = ρ Adg (ω)Γ}
is right (SO(n) × Rn )–invariant, so the Chaplygin sphere is a LR system on the direct product SO(n) × Rn . If we take the fixed orthonormal base E1 , . . . , En such that Γ = En , then the constraint (6) takes the form r˙i = Wi = ρΩin ,
i = 1, . . . , n − 1,
r˙n = Wn = 0,
where Ωij = hΩ, Ei ∧ Ej i. The last constraint is holonomic, and for the physical motion we take rn = ρ. From now on we take SO(n) × Rn−1 for the configuration space of the rolling sphere, where Rn−1 is identified with the affine hyperplane ρΓ + H. The Chaplygin sphere is Rn−1 -Chaplygin system and the reduced space D/Rn−1 is the tangent bundle T SO(n).
ˇ ´ BOZIDAR JOVANOVIC
4
Remark 1. Together to the Chaplygin sphere we can also consider the rubber Chaplygin sphere, defined as a system (4), (6) subjected to the additional right-invariant constraints Ωij = 0, 1 ≤ i < j ≤ n − 1 describing the no-twist condition at the contact point [11, 21]. 2.2. Dynamics. From the constraints (6) we find the form of reaction forces in the right-trivialization in which the equations (2) read
(9)
M˙ = −ρΛ ∧ Γ, ˙ = Λ, mW
(10)
g˙ = Ω · g,
(11)
r˙ = W.
(8)
where M = Adg (Iω) is the ball angular momentum in the space and Λ ∈ Rn is the Lagrange multiplier. ˙ On the other Differentiating the constraints (6) and using (9) we get Λ = mρΩΓ. hand ˙ ∧ Γ = mρ Ω˙ Γ ⊗ Γ + Γ ⊗ Γ Ω ˙ = mρ prh (Ω), ˙ (12) Λ ∧ Γ = mρ(ΩΓ)
where h ⊂ so(n) is the linear subspace
h = Rn ∧ Γ,
(13)
and prh : so(n) → h, prh (ξ) = (ξΓ) ∧ Γ = ξΓ ⊗ Γ + Γ ⊗ Γξ is the orthogonal projection with respect to the scalar product (5). Whence, (8), (10) is a closed system on T SO(n), representing the Chaplygin reduction of Rn−1 symmetry. Now we need to write it in the left trivialization of T SO(n). Let γ = g −1 Γ be the vertical vector in the frame attached to the ball. Then (14)
Adg−1 (h) = Rn ∧ γ =: hγ .
˙ and the relations (12) and pr γ (ξ) = (ξ ·γ)∧γ = From the identity ω˙ = Adg−1 (Ω) h ξ γ ⊗ γ + γ ⊗ γ ξ we get I ω˙ = [Iω, ω] − mρ2 (ω˙ γ ⊗ γ + γ ⊗ γ ω). ˙ Let us denote mρ2 by D and let (15)
k = Iω + D prhγ ω = Iω + D(ω γ ⊗ γ + γ ⊗ γ ω) ∈ so(n)∗
be the angular momentum of the ball relative to the contact point (see [16]). By using the Poisson equation (16)
γ˙ = −ωγ
d it easily follows dt (ωγ ⊗ γ + γ ⊗ γω) = ωγ ˙ ⊗ γ + γ ⊗ γ ω˙ + [ωγ ⊗ γ + γ ⊗ γω, ω]. Therefore, the reduced Chaplygin sphere equations, in variables (ω, g) are given by
(17)
k˙ = [k, ω],
(18)
g˙ = g · ω,
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
5
while the reduced kinetic energy is given by Lred = 12 hk, ωi. The system is additionally left SO(n−1)–invariant where the action of SO(n−1) is given by the rotations around the vertical vector Γ. The closed system (16), (17) in variables (ω, γ) represents the reduction of SO(n − 1)–symmetry to so(n) × S n−1 ∼ = (T SO(n))/SO(n − 1). It possesses an invariant measure with density [16] q (19) µ = det(I + D prhγ ).
We can also consider the system (17), (18) on the cotangent bundle T ∗ SO(n) in ∗ n−1 variables (k, g) or in coordinates (k, γ) on . Then qthe reduced space so(n) × S
the invariant measure is given by µ = 1/ det(I + D prhγ ) (see [13]). In the case n = 3, under the isomorphism between so(3) and R3
(20)
ωij = εijl ωl ,
kij = εijl kl ,
from (17) and (16) we obtain the classical Chaplygin’s ball equations ~k˙ = ~k × ~ω , (21) ~γ˙ = ~γ × ~ω , where ~k = I~ ω + D~ ω − D(~ ω , ~γ )~γ and (22)
I = diag(I1 , I2 , I3 )
is the inertia operator of the ball. In the space (~ω, ~γ ) the density of an invariant measure (19), up to the multiplication by a constant factor is equal to p 1 − D(γ, (I + DI)−1 γ), the expression given by Chaplygin in [8]. 3. Reduced System in Redundant Coordinates
3.1. Reduction to T ∗ S n−1 . From (8) we have d d (23) (prk M ) = (pr Adg (Iω)) dt dt k d d (pr Adg (Iω + D prhγ ω)) = (prk Adg k) = 0, = dt k dt d d (prkγ (Iω)) = dt (prkγ k) = 0, where k ∼ or, equivalently, dt = so(n − 1) is orthogonal complement to h with respect to (5) and kγ := Adg−1 k = (Rn ∧ γ)⊥ . The integral (23) is actually the momentum mapping Φ : T ∗ SO(n) → so(n − 1)∗ ∼ = k,
Φ(k, g) = prk Adg (k)
of the left SO(n − 1)–action. Namely, since for ξ ∈ so(n − 1) the associated vector field ξSO(n)×Rn−1 is a section of (7), we have the Noether conservation low (3). The integral is Rn−1 –invariant and descend to T ∗ SO(n). For n = 3 we have the classical area integral (~k, ~ ω). As in the usual symplectic reduction, we can use the momentum mapping Φ to reduce the system to Mη = Φ−1 (η)/SO(n − 1)η , where SO(n)η is the isotropy
ˇ ´ BOZIDAR JOVANOVIC
6
group of η ∈ so(n − 1)∗ (e.g., see [19]). The reduced space Mη is the Oη bundle over T ∗ S n−1 . Here Oη is the coadjoint orbit of η. We shall consider the simplest but still very interesting case, when we fix the value of the momentum mapping Φ to be zero (24)
prkγ (Iω) = prkγ k = 0.
Whence, both k and Iω belong to (14). Now, let us introduce new variables p, ξ ∈ Rn orthogonal to γ (25)
(γ, p) = (γ, ξ) = 0,
such that k = γ ∧ p,
(26)
ω = I −1 (γ ∧ ξ).
Lemma 1. The variables p and ξ are related via p = ξ − DI −1 (γ ∧ ξ)γ
(27)
Proof. The proof directly follows from the relations (15) and (26). From (27), under the conditions (25), the variable ξ can be uniquely expressed via p and γ. Note that the coordinates (γ, p) can be considered as redundant coordinates of the cotangent bundle of the sphere T ∗ S n−1 realized as a subvariety of R2n defined by constraints φ1 ≡ (γ, γ) = 1,
(28)
φ2 ≡ (γ, p) = 0.
Theorem 2. The reduced Chaplygin–sphere problem on T ∗ S n−1 = Φ−1 (0)/SO(n− 1) is described by the equations (29)
γ˙ = Xγ (γ, p) = −ωγ = −I −1 (γ ∧ ξ(γ, p))γ
(30)
p˙ = Xp (γ, p) = −ωp = −I −1 (γ ∧ ξ(γ, p))p
The equations (29) and (30) preserve the reduced Hamiltonian function 1 1 H(γ, p) = hk, ωi = hγ ∧ p, I −1 (γ ∧ ξ(γ, p))i 2 2 (which is now unique only on the subvariety (28)) and the reduced momentum K(γ, p) = hγ ∧ p, γ ∧ pi = (γ, γ)(p, p) − (γ, p)2 = (p, p).
(31)
Proof of Theorem 2. The equation (29) follows directly from the Poisson equation (16). On the other hand, from the equation (17) we get γ˙ ∧ p + γ ∧ p˙ = [γ ∧ p, ω] =⇒
−ωγpT − p(−ωγ)T + γ ∧ p˙ = γpT ω − pγ T ω − ωγpT + ωpγ T
=⇒
γ ∧ p˙ = ωpγ T + γpT ω = (ωp) ∧ γ
=⇒
p˙ = −ωp + λγ.
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
7
The multiplier λ is equal to zero. Indeed, from (28) we have d φ2 = (γ, ˙ p) + (γ, p) ˙ = (−ωγ, p) + (γ, −ωp) + λ(γ, γ) = λ = 0. dt 3.2. Chaplygin Reducing Multiplier. The reduction of the Chaplygin–sphere problem to T ∗ S n−1 is not a Chaplygin reduction. Nevertheless, the idea of a time reparametrization in order to transform the system to the Hamiltonian form is still applicable. The vector field X = (Xγ , Xp ) of the system (29), (30) can be written in the almost Hamiltonian form iX (w) = dH, where the form w is a non-degenerate 2-form on T ∗ S n−1 , a semi-basic perturbation of the symplectic form ω = (dp1 ∧ dγ1 + · · · + dpn ∧ dγn )|T ∗ S n−1
(32)
(see [19]). Let w be a nondegenerate 2-form on an even dimensional manifold M . For a differential equation x˙ = X that can be written in the almost Hamiltonian form iX w = dH, the Chaplygin multiplier is a nonvanishing function N such that ω ˜ = 1 ˜ N w is closed. Since iX˜ ω ˜ = dH, X = N X, after the time substitution dτ = N dt, d ˜ with respect to the x=X the system x˙ = X becomes the Hamiltonian system dτ symplectic form ω ˜ [24, 7, 25, 11]. More generally, N is the Chaplygin multiplier if there exist a 2-form w0 such that iX w0 = 0 and ω ˜ = N (w − w0 ) is symplectic (see [11]). Then, as above, the system x˙ = X becomes the Hamiltonian system d ˜ ˜. dτ x = X with respect to the symplectic form ω Alternatively, a transparent and classical way to introduce the Chaplygin reducing multiplier for our system is as follows. Let N (γ) be a differentiable nonvanishing positive function in a neighborhood of S n−1 . Consider the coordinate transformation (γ, p) 7−→ (γ, p˜),
p˜ = N p
defined is some neighborhood of T ∗ S n−1 and the symplectic form (33)
ω ˜
= d˜ p1 ∧ dγ1 + · · · + d˜ pn ∧ dγn |T ∗ S n−1 = N (ω + d ln N ∧ dγ1 + . . . d ln N ∧ dγ n )|T ∗ S n−1 .
Then N is a Chaplygin multiplier for the reduced system if the equations (29), (30) in the new time dτ = N (q)dt becomes Hamiltonian with respect to the form ω ˜ . If N is a Chaplygin multiplier then from the Liouville theorem we have (34)
LX˜ (˜ ω n−1 ) = 0
⇐⇒
LX (N n−2 ω n−1 ) = 0,
i.e., the original system has the invariant measure with density N (γ)n−2 .
ˇ ´ BOZIDAR JOVANOVIC
8
3.3. Homogeneous Sphere. In [19], it is proved that the equations of motion of the homogeneous ball on the reduced space Mη = Φ−1 (η)/SO(n−1)η , for any value of the SO(n − 1) momentum mapping, are already hamiltonian. This interesting result, for the motion on T ∗ S n−1 = Φ−1 (0)/SO(n − 1) can be easily derived from Theorem 2. Suppose the inertia operator I is multiplication by a constant ρ > 0. Then the ρ equation (27), under the conditions (25), gives ξ = ρ+D p. The reduced system (29), (30) takes the form γ˙ =
1 p, ρ+D
p˙ = −
(p, p) γ, ρ+D
representing the geodesic flow of the metric of the round sphere multiplied by ρ+D. 4. Hamiltonization In this section we shall perform the Hamiltonization of the reduced Chaplygin sphere (29), (30) for the inertia operator defined on the base Ei ∧ Ej via (35)
I(Ei ∧ Ej ) =
ai aj D Ei ∧ Ej , D − ai aj
1 ≤ i < j ≤ n,
where 0 < ai aj < D, 1 ≤ i, j ≤ n. The form of the inertia operator as well as the form of the Chaplygin multiplier below is motivated by the corresponding formulas in the problem of motion of the n-dimensional Veselova problem as well as the rubber Chaplygin ball given in [17] and [21], respectively. Let A = diag(a1 , . . . , an ). Theorem 3. The reduced Chaplygin sphere equations (29), (30), defined by the inertia tensor (35), read 1 (p, γ) (γ, Ap) (γ, γ) p− A−1 γ + 2 γ− 2 Ap, D D(γ, A−1 γ) D (γ, A−1 γ) D (γ, A−1 γ) (p, A−1 γ) (p, p) (p, Ap) (p, γ) (37) p˙ = p− A−1 γ + 2 γ− 2 Ap. D(γ, A−1 γ) D(γ, A−1 γ) D (γ, A−1 γ) D (γ, A−1 γ)
(36) γ˙ =
Proof. From the definition (35), the angular velocity is given by (38)
ω = I −1 (γ ∧ ξ) = A−1 γ ∧ A−1 ξ −
1 γ ∧ ξ. D
Now, the equation (27), under the conditions (25), can be solved 1 (Ap − (p, Aγ)γ) . D(γ, A−1 γ) Thus ω = A−1 γ ∧ p − γ ∧ Ap/D /D(γ, A−1 γ) and (36), (37) simply follows from (29), (30). (39)
ξ=
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
9
4.1. Invariant Measure. Note that the reduced Hamiltonian 1 1 hγ ∧ p, A−1 γ ∧ p − γ ∧ Api (40) H(γ, p) = −1 2D(γ, A γ) D as well as the system (36), (37) itself, is defined on ˆ 2n = R2n \ {γ = 0}. R
(41)
ˆ 2n the system preserve functions φ1 and φ2 . Also, considered on R Consider the standard spherical coordinates (θ, r) = (θ1 , . . . , θn−1 , r) in Rn (γ) and the corresponding canonical momenta (πθ , πr ) = (π1 , . . . , πn−1 , πr ) in R2n (γ, p) with respect to the canonical symplectic form: dp1 ∧ dγ1 + · · · + dpn ∧ dγn = dπ1 ∧ dθ1 + · · · + dπn−1 ∧ dθn−1 + dπr ∧ dr. Then the volume form in R2n can be represented as (42) Ω = dp1 ∧ dγ1 ∧ · · · ∧ dpn ∧ dγn = (dπ1 ∧ dθ1 ∧ · · · ∧ dπn−1 ∧ dθn−1 ) ∧ dpr ∧ dr, p p where r = (γ, γ) and pr = (γ, p)/ (γ, γ). The coordinates (θ, πθ ) are canonical coordinates (the symplectic form (32) equals dθ1 + · · · + dπn−1 ∧ dθn−1 ) and σ = dπ1 ∧ dθ1 ∧ · · · ∧ dπn−1 ∧ dθn−1 =
1 ω n−1 (n − 1)!
is the canonical volume form on the cotangent bundle T ∗ S n−1 , naturally extended ˆ 2n . to R For example, in the case n = 3, the canonical transformation (γ, p) 7→ (θ, r, πθ , πr ) is given by: γ1
= r cos θ1 ,
γ2
= r sin θ1 cos θ2 ,
γ3
= r sin θ1 sin θ2 ,
p1
= π1 sin θ1 /r + πr cos θ1 ,
p2
= −π1 cos θ1 cos θ2 /r + π2 sin θ2 /(r sin θ1 ) + πr sin θ1 cos θ2
p3
= −π1 cos θ1 sin θ2 − π2 cos θ2 /(r sin θ1 ) + πr sin θ1 sin θ2 .
Proposition 4. The reduced Chaplygin system (36), (37) possesses an invariant measure µ(γ) σ = (A−1 γ, γ)−(n−2)/2 σ .
(43)
ˆ 2n is Proof. The divergence of the vector field X in R n X ∂ γ˙ i ∂ p˙ i (γ, Ap) (γ, A−1 p) (44) div(X) = + Ψ, = (n − 2) + + ∂γi ∂pi D(γ, A−1 γ) D2 (γ, A−1 γ) i=1
where
Ψ=
2(γ, γ) trA−1 trA 2(A−2 γ, γ) + 2 − − ) (γ, p). D(γ, A−1 γ)2 D (γ, A−1 γ)2 D(γ, A−1 γ) D2 (γ, A−1 γ)
ˇ ´ BOZIDAR JOVANOVIC
10
Whence, on the invariant submanifold φ2 = πr = 0, in the view of (36), we get n X ∂ γ˙ i (γ, A−1 γ) ˙ ∂ p˙ i µ˙ = (n − 2) + =− . −1 ∂γi ∂pi (γ, A γ) µ i=1
In other words, the density µ(γ) satisfies the Liouville equation n n X X ∂ q˙i ∂ p˙ i ∂µ =0 +µ + q˙i (45) div(µX) = ∂qi ∂qi ∂pi i=1 i=1
on the manifold φ2 = πr = 0. On the other side, from (42) we obtain
LX (µΩ) = LX (µσ) ∧ dπr ∧ dr + µσ ∧ LX (dπr ∧ dr).
(46)
Since the functions φ1 , φ2 are invariants of the vector field X, the Lie derivatives LX dπr and LX dr equal zero. Further, (45) implies that the left hand side of (46) is also equal to zero on the invariant subvariety φ2 = πr = 0. Thus we conclude LX (µσ)|T ∗ S n−1 = 0 as required. 4.2. Time Reparametrization. According to the constraints (28), instead of (40) we can use the Hamiltonian function 1 (47) H(γ, p) = D(γ, A−1 γ)(p, p) − (p, Ap) 2D2 (γ, A−1 γ)
As follows from Proposition 4 and (34), if the reduced Chaplygin system on T S n−1 is transformable to a Hamiltonian form by a time reparameterization, then p the corresponding reducing multiplier N should be proportional to 1/ (γ, A−1 γ). ∗
Theorem 5. Under the time substitution (48)
dτ = N dt =
D
and an appropriate change of momenta (49)
(γ, p) 7−→ (γ, p˜),
1 p dt (A−1 γ, γ) p˜ =
D
p
1 p (γ, A−1 γ)
the reduced system (36), (37) becomes a Hamiltonian system describing a geodesic flow on S n−1 with the Hamiltonian 1 (50) H(γ, p˜) = D(γ, A−1 γ)(˜ p, p˜) − (˜ p, A˜ p) . 2
Proof. Consider the cotangent bundle T ∗ S n−1 realized as a submanifold of R2n given by (51)
ψ1 ≡ (γ, γ) = 1,
ψ2 ≡ (γ, p˜) = 0.
The canonical Poisson bracket on T ∗ S n−1 with respect to the symplectic form (33) can be described by the use of the Dirac bracket (see [10, 22, 1]): {F, G}d = {F, G} − ({F, ψ1 }{G, ψ2 } − {F, ψ2 }{G, ψ1 })/{ψ1 , ψ2 },
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
11
where {F, G} =
n X ∂F ∂G ∂F ∂G . − ∂γi ∂ p˜i ∂ p˜i ∂γi i=1
ˆ 2n , the bracket {·, ·}d is degenerate and has two Casimir funcConsidered on R tions ψ1 and ψ2 . The symplectic leaf given by (51) is exactly the cotangent bundle T ∗ S n−1 endowed with the canonical symplectic form. Under the mapping (49), the Hamiltonain (47) transforms to (50). With the above notation, the geodesic flow defined by Hamiltonian function (50), in the time τ , is the restriction to (51) of (52)
γi′ =
d γi = {γi , H}d , dτ
p˜′i =
d p˜i = {˜ pi , H}d , dτ
i = 1, . . . , n.
It is convenient to find equations (52) by using the Lagrange multipliers (see [22, 1]). Introduce H ∗ = H − λψ1 − µψ2 . The equations (52) are then given by ∂H ∗ ∂H = − µ˜ p = D(A−1 γ, γ)˜ p − A˜ p − µγ, ∂ p˜ ∂ p˜ ∂H ∗ ∂H p˜′ = − =− + λγ + µ = −D(˜ p, p˜)A−1 γ + λγ + µ˜ p ∂γ ∂γ
γ′ =
where multipliers λ and µ are determined from the condition that the constraint functions ψ1 and ψ2 are integrals of the motion. The straightforward calculations yield (γ, A˜ p) γ, (γ, γ) (˜ p, A˜ p) (γ, A˜ p) p˜′ = −D(˜ p, p˜)A−1 γ + γ− p˜. (γ, γ) (γ, γ) γ ′ = D(A−1 γ, γ)˜ p − A˜ p+
(53) (54)
In the time t, after inverting the mapping (49), the equation (53) takes the form p (γ, Ap) 1 −1 −1 γ , γ˙ · D (γ, A γ) = p D(A γ, γ)p − Ap + (γ, γ) D (γ, A−1 γ)
i.e.,
(55)
γ˙ =
1 1 (γ, Ap) p− 2 Ap + 2 γ, D D (γ, A−1 γ) D (γ, A−1 γ)(γ, γ)
which coincides with (36) on the points of T ∗ S n−1 . Further, ! ! p p p d d d p p p˜ = = D (γ, A−1 γ) = dτ dτ D (γ, A−1 γ) dt D (γ, A−1 γ) ! p 1 (A−1 γ, γ) ˙ 1 d (56) (γ, A−1 γ) = p˙ − p + p˙ p = p p −1 −1 −1 dt (γ, A γ) (γ, A γ) (γ, A γ)
ˇ ´ BOZIDAR JOVANOVIC
12
Finally, subtracting (49) in the left hand side of (54), combining with (55) and (56), we get (57) p˙
=
(p, Ap) (γ, Ap) (p, p) A−1 γ + 2 γ− 2 p −1 −1 D(γ, A γ) D (γ, A γ)(γ, γ) D (γ, A−1 γ)(γ, γ) (γ, Ap) (γ, p) (p, A−1 γ) p+ 2 + p− 2 p −1 −1 2 D(γ, A γ) D (γ, A γ) D (γ, A−1 γ)(γ, γ) −
As above, the equations (37) and (57) are different, but they coincide on the invariant manifold φ1 = ψ1 = 1, φ2 = ψ2 = 0. The theorem is proved. Remark 2. After the isomorphism (20), the operator (35) defines the inertia tensor (22) by I1 = a2 a3 D/(D − a2 a3 ), I2 = a3 a1 D/(D − a3 a1 ), I3 = a2 a3 D/(D − a2 a3 ). In the other direction, given an inertia tensor (22), the matrix A = diag(a1 , a2 , a3 ) is determined via p p i = 1, 2, 3. ai = I1 I2 I3 D(Ii + D)/Ii (I1 + D)(I2 + D)(I3 + D), p The Chaplygin reducing multiplier 1/ 1 − D(γ, (I + DI)−1 γ) given in [6], up to a p multiplication by a constant, coincides with 1/D (A−1 γ, γ). 5. Integrability
5.1. Chaplygin Sphere and the Veselova Problem. The momentum integral (31) after the mapping (49) becomes the integral (58)
K(γ, p˜) = D2 (A−1 γ, γ)(˜ p, p˜)
of the geodesic flow (53), (54). Let 1 (A˜ p, p˜) . 2 Since H = K/2D − G, the quadratic function G is also the integral of the flow. On the other side, (59) is proportional to the Hamiltonian of the reduced multidimensional nonholonomic Veselova problem [17]. More precisely, for the inertia operator defined by
(59)
(60)
G=
−1 I(Ei ∧ Ej ) = a−1 i aj Ei ∧ Ej ,
1 ≤ i < j ≤ n,
the Veselova problem is reducible to T ∗ S n−1 . In redundant coordinates (γ, p) the reduced system is given by (see [17, 18]) 1 1 γ˙ = (−(p, Aγ)γ + (γ, γ)Ap) = (−(p, Aγ)γ + Ap) , (γ, A−1 γ) (γ, A−1 γ) (p, Ap) 1 (−(p, Ap)γ + (p, γ)Ap) = γ. p˙ = − (γ, A−1 γ) (γ, A−1 γ) Furthermore, as it follows from [17, 18], under the time substitution (48) and the change of momenta (49) the reduced system becomes a Hamiltonian system describing a geodesic flow on S n−1 with the Hamiltonian H = D2 G. Let us suppose ai 6= aj , i 6= j. Then the Hamiltonian (59) has the St¨ackel form in sphero-conical variables and the geodesic flow on S n−1 determined by H
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
13
is completely integrable (see [17]). Whence the system (53), (54) is completely integrable as well. Therefore, as in the three-dimensional case [12], we have Theorem 6. The reduced multidimensional nonholonomic Chaplygin–sphere problem defined by inertia operator (35) and the reduced Veselova problem defined by inertia operator (60) have the same invariant toric foliation. Let T be a regular, (n−1)-dimensional invariant torus. Then there exist angle coordinates ϕ1 , . . . , ϕn−1 on T in which both problems simultaneously take the form ϕ˙ 1 = N (ϕ1 , . . . , ϕn−1 ) ω1 , . . . , ϕ˙ n−1 = N (ϕ1 , . . . , ϕn−1 ) ωn−1 with different frequencies ω1 , . . . , ωn−1 . Remark 3. Let x(t) be a geodesic line on the ellipsoid E n−1 = {x = (x1 , . . . , xn ) ∈ Rn | (x, Ax) = 1} endowed with the standard metric. Then, up to the time p rescaling, the unit normal vector γ(t) = Ax(t)/ (Ax(t), Ax(t)) is a solution of the reduced generalized Veselova system defined by inertia tensor (60) (see [18]). In this sense, the momentum integral (58) corresponds to classical Joachimsthal’s integral of the geodesic flow on the ellipsoid E n−1 (see [22]). 5.2. Lagrange Case. The system is integrable even if not all ai are distinct. For any pair of equal parameters ai = aj , the geodesic flow (53), (54) has the additional linear integral fij = γi p˜j − γj p˜i . For example, let n = 4 and a1 = a2 6= a3 = a4 . Then the complete set of commuting integrals is f12 , f34 and H. If we have at least three equal parameters, the system is integrable according to the non-commutative version of the Liouville theorem. We will consider the case a1 = a2 = · · · = an−1 6= an .
(61)
Namely, in general, for n ≥ 4, the operator (35) is not a physical inertia operator of a multidimensional rigid body (see [16]). However, by taking conditions (61) and 2an D > a1 an + a1 D, we get the operator Iω = Jω + ωJ, where J = diag(J1 , J1 , . . . , J1 , Jn ), J1 =
a21 D , 2(D − a21 )
Jn =
a21 D a1 an D − , D − a1 an 2(D − a21 )
representing a SO(n − 1)–symmetric rigid body (multidimensional Lagrange case [2]) with a mass tensor J. Due to the additional SO(n − 1)-symmetry, the geodesic flow (53), (54) has the integrals fij , 1 ≤ i < j ≤ n − 1. Thus, in the original coordinates we get integrals (62)
Fij = (γ, A−1 γ)(γi pj − γj pi )2 ,
1 ≤ i < j ≤ n − 1.
In this case we do not need Hamiltonization to integrate the reduced system, it is already integrable according to the Euler–Jacobi theorem (e.g., see [1]). Since the
14
ˇ ´ BOZIDAR JOVANOVIC
generic invariant manifolds given by H and integrals (62) are two-dimensional and the system has an invariant measure we have [1]: Theorem 7. The Lagrange case of the reduced Chaplygin system (36), (37) is solvable by quadratures. Remark 4. Although the Lagrangian (4) is additionally invariant with respect to the right SO(n − 1)–action, the integrals (62) are not Noether’s integrals. The reason is that the associated vector fields are not sections of the distribution (7). For n = 3 and I1 = I2 , the corresponding integral of the system (21) is F = k32 − D(γ, (I + DI)−1 γ)k32 . Acknowledgments. I am greatly thankful to Yuri N. Fedorov for useful discussions. The research was supported by the Serbian Ministry of Science, Project 144014, Geometry and Topology of Manifolds and Integrable Dynamical Systems. References [1] Arnold V I, Kozlov V V, Neishtadt A I 1985 Mathematical aspects of classical and celestial mechanics. Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamental’nye Napravleniya, Vol. 3, VINITI, Moscow 1985. English transl.: Encyclopadia of Math. Sciences, Vol.3, SpringerVerlag, Berlin 1989. [2] Beljaev A V 1981 Motion of a multidimensional rigid body with a fixed point in a gravitational force field Mat. Sb. 114(156) no. 3, 465-470 (Russian). [3] Bloch A M, Krishnaprasad P S, Marsden J E, Murray R M 1996 Nonholonomical mechanical systems with symmetry Arch. Rational Mech. Anal. 136 21-99. [4] Bloch A M, Zenkov D V 2003 Invariant Measures of Nonholonomic Flows With Internal Degrees of Freedom. Nonlinearity 16, 1793–1807. [5] Borisov A, Mamaev I 2001 Chaplygin’s ball rolling problem is Hamiltonian. (Russian) Mat. Zametki 70, no. 5, 793–795; English translation: Math. Notes 70 (2001), no. 5-6, 720–723 [6] Borisov A, Mamaev I 2007 Isomprphism and Hamiltonizations of Some Nonholonomic Systems, Sib. Mat. Zh. 46, no. 1, 33-45. (see also arXiv:nlin/0509036) [7] Cantrijn F, Cortes J, de Leon M, Martin de Diego D 2002 On the geometry of generalized Chaplygin systems. Math. Proc. Cambridge Philos. Soc. 132 no. 2, 323-351; arXiv: math.DS/0008141. [8] Chaplygin S A 1903 On a rolling sphere on a horizontal plane. Mat. Sbornik 24 139-168 (Russian) [9] Chaplygin S A 1911 On the theory of the motion of nonholonomic systems. Theorem on the reducing multiplier. Mat. Sbornik 28 no. 2, 303-314 (Russian). [10] Dirac P A 1950 On generalized Hamiltonian dynamics. Can. J. Math. 2, no.2, 129–148. [11] Ehlers K, Koiller J, Montgomery R, Rios P 2005 Nonholonomic systems via moving frames: Cartan’s equivalence and Chaplygin Hamiltonization, The breadth of symplectic and Poisson geometry, 75–120, Progr. Math., 232, Birkhuser Boston, Boston, MA, arXiv: math-ph/0408005. [12] Fedorov Yu 1989 Two Integrable Nonholonomic Systems in Classical Dynamics, Vest. Moskov. Univ. Ser I Mat. Mekh. no 4, 38–41 (Russian). [13] Fedorov Yu 1996 Dynamical systems with an invariant measure on the Riemannian symmetric pairs (GL(N ), SO(N )). (Russian) Reg. Ch. Dyn. 1, no. 1, 38–44. [14] Fedorov Yu 2007 A Discretization of the Nonholonomic Chaplygin Sphere Problem, SIGMA 3 (2007), 044, 15 pages, arXiv:nlin/0612037 [15] Fedorov Yu 2009, A Complete Complex Solution of the Nonholonomic Chaplygin Sphere Problem, preprint [16] Fedorov Yu N, Kozlov V V 1995 Various aspects of n-dimensional rigid body dynamics Amer. Math. Soc. Transl . Series 2, 168 141–171.
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
15
[17] Fedorov Yu N, Jovanovi´ c B 2004 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces. J. Non. Sci., 14, 341-381, arXiv: math-ph/0307016. [18] Fedorov Yu N, Jovanovi´ c B 2009 On the Hamiltonization of the Generalized Veselova LR System, preprint [19] Hochgerner S, Garc´ıa-Naranjo L 2008 An (almost) symplectic view of Chaplygin ball, arXiv:08105454 [math-ph] [20] Koiller J 1992 Reduction of some classical non-holonomic systems with symmetry Arch. Rational Mech. 118 113-148. [21] Jovanovi´ c B 2009 LR and L+R systems, arXiv: 0902.1656 [math-ph] [22] Moser J 1980 Geometry of quadrics and spectral theory. Chern Symposium 1979, Berkeley, Springer, 147–188. [23] Schneider D. 2002 Nonholonomic Euler-Poincar´ e Equations and Stability in Chaplygin’s Sphere. Dynamical Systems: An International Journal., 17 No. 2, 87–130 [24] Stanchenko S 1989 Nonholonomic Chaplygin systems. Prikl.Mat.Mekh. 53, no.1, 16–23. English transl.: J.Appl.Math.Mech. 1989 53, no.1, 11–17. [25] Tatarinov Ya 1990 Nonholonomic Systems in Comparatione to Hamiltonian Ones, Doctoral Dissertation, Moscow State University, Moscow. [26] Veselov A P, Veselova L E 1988 Integrable nonholonomic systems on Lie groups Mat. zametki 44 no. 5, 604-619 (Russian); English translation: 1988 Mat. Notes 44 no. 5. Mathematical Institute SANU, Kneza Mihaila 36, 11000, Belgrade, Serbia E-mail address: [email protected]
HAMILTONIZATION AND INTEGRABILITY OF THE CHAPLYGIN SPHERE IN Rn ˇ ´ BOZIDAR JOVANOVIC
Abstract. We prove that the n-dimensional Chaplygin–sphere problem for the zero value of the SO(n − 1)–momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.
1. Introduction Nonholomic systems are not Hamiltonian. Apparently, Chaplygin was one of the first who considered a time reparametrization in order to transform nonholonomic systems to the Hamiltonian form [9]. Also, after [8], one of the most famous solvable problems in nonholonomic mechanics, describing the rolling without slipping of a balanced ball over a horizontal surface, is refered as the Chaplygin sphere, see [1, 15]. It is interesting that the Hamiltonization of the system by the use of a time reparametrization is done just recently by Borisov and Mamaev [5, 6]. Fedorov and Kozlov constructed natural n-dimensional model of the Chaplygin– sphere problem and found an invariant measure [16]. Various aspects of the problem are studied in [23, 14, 19]. In [19], it is proved that the reduced equations of motion of the homogeneous ball are already Hamiltonian. However, the general problem of integrability and Hamiltonization is still unsolved. 1.1. Natural Nonholonomic Systems. Let Q be a n–dimensional Riemannian manifold with a nondegenerate metric κ(·, ·), V : Q → R be a smooth function and let D be a nonintegrable (n − k)–dimensional distribution of the tangent bundle T Q. A smooth path q(t) ∈ Q, t ∈ ∆ is called admissible (or allowed by constraints) if the velocity q(t) ˙ belongs to Dq(t) for all t ∈ ∆. Let q = (q1 , . . . , qn ) be some local coordinates on Q in which the constraints are written in the form n X αji q˙i = 0, j = 1, . . . , k, (1) (αjq , q) ˙ = i=1
where αj are independent 1-forms. The admissible path q(t) is a motion of the natural mechanical nonholonomic system (Q, κ, V, D) (or a nonholonomic geodesic for V ≡ 0) if it satisfies the Lagrange–d’Alambert equations k
(2)
d ∂L ∂L X λj αj (q)i , = + dt ∂ q˙i ∂qi j=1
MSC: 37J60, 37J35, 70H45 1
i = 1, . . . , n.
ˇ ´ BOZIDAR JOVANOVIC
2
Here the Lagrange multipliers λj are chosen such that the solutions q(t) satisfy constraints (1) and the Lagrangian is given by the difference of the kinetic and Pk P potential energy: L(q, q) ˙ = 12 ij κij q˙i q˙j − V (q). The expression j=1 λj αj (q)i represents the reaction forces of the constraints (1). P After the Legendre transformation pi = ∂L/q˙i = j κij q˙j one can also write the Lagrange-d’Alambert equations as a first-order system on the cotangent bundle k
∂H X λj αj (q)i , i = 1, . . . , n, + ∂qi j=1 P where the Hamiltonian is H(q, p) = 21 ij κij pi pj + V (q). As for the Hamiltonian systems, it is the first integral of the system. q˙i =
∂H , ∂pi
p˙ i =
1.2. Symmetries. Suppose that a Lie group K acts by isometries on (Q, κ) preserving the potential function V (the Lagrangian L is K- invariant) and let ξQ be the vector field on Q associated to the action of one-parameter subgroup exp(tξ), ξ ∈ k = TId K. The following version of the Noether theorem holds (see [1, 3]): if ξQ is a section of the distribution D then d ∂L d (3) , ξQ = (p, ξQ ) = 0. dt ∂ q˙ dt
On the other side, let ξQ be transversal to D, for all ξ ∈ k. In addition, suppose that Q has a principal bundle structure π : Q → Q/K and that D is the collection of horizontal spaces of a principal connection. Then (Q, κ, V, D) is called a KChaplygin system. The system (2) is K-invariant and reduces to the tangent bundle T (Q/K) = D/K (for the details see [20, 3, 7, 25]). In some cases the equations (2) have a rather strong property – an invariant measure (e.g, see [1, 4]). Within the class of K-Chaplygin systems, the existence of an invariant measure is closely related with their reduction to a Hamiltonian form after an appropriate time rescaling dτ = N dt (see [9, 24, 17, 7, 25, 11]). Nonholonomic systems on unimodular Lie groups with right-invariant constraints and left-invariant metrics, so called LR systems, always have an invariant measure [26]. Recently, a nontrivial example of a nonholonomic LR system on the group SO(n) (n-dimensional Veselova problem), which can be regarded also as a SO(n − 1)–Chaplygin system such that the reduced system on S n−1 = SO(n)/SO(n − 1) is Hamiltonian after the time rescaling, is given in [17].
1.3. Outline and Results of the Paper. In Section 2, we recall the equations of motion of the Chaplygin Sphere. The reduction of the system to the cotangent bundle of the sphere T ∗ S n−1 for a zero value of the SO(n − 1) momentum mapping is described in Section 3. The calculation of an invariant measure as well as the time reparametrization and the reduction of the system to the Hamiltonian form for a specific choice of an inertia operator of the ball is given in Section 4. It appears that the obtained Hamiltonian system is an integrable geodesic flow. Moreover, as in the 3-dimensional case [12],
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
3
the reduced system is closely related to the reduced nonholonomic Veselova problem (Section 5). 2. Chaplygin Sphere 2.1. Cinematics. Following [16, 14], consider the Chaplygin–sphere problem of rolling without slipping of an n-dimensional balanced ball (the mass center C coincides with the geometrical center) of radius ρ on an (n − 1)-dimensional hyperspace H in Rn . For the configuration space we take the direct product of Lie groups SO(n) and Rn , where g ∈ SO(n) is the rotation matrix of the sphere (mapping frame attached to the body to the space frame) and r ∈ Rn is the position vector of its center C (in the space frame). For a trajectory (g(t), r(t)) define angular velocities of the sphere in the moving and the fixed frame as well as the velocity in the fixed frame by ω = g −1 g, ˙ Ω = gg ˙ −1 , W = r. ˙ The Lagrangian of the system is then given by 1 1 (4) L = hIω, ωi + m(W, W). 2 2 Here I : so(n) → so(n) and m are the inertia tensor and mass of the ball, h·, ·i is given by 1 (5) hX, Y i = − tr(XY ), 2 and (·, ·) is the Euclidean scalar product. Let Γ ∈ Rn be a vertical unit vector (considered in the fixed frame) orthogonal to the hyperplane H and directed from H to the center C. The condition for the sphere to role without slipping leads that the velocity of the contact point is equal to zero: W − ρΩΓ = 0 .
(6) The distribution (7)
D = {(g, r, ω, W) | W = ρ Adg (ω)Γ}
is right (SO(n) × Rn )–invariant, so the Chaplygin sphere is a LR system on the direct product SO(n) × Rn . If we take the fixed orthonormal base E1 , . . . , En such that Γ = En , then the constraint (6) takes the form r˙i = Wi = ρΩin ,
i = 1, . . . , n − 1,
r˙n = Wn = 0,
where Ωij = hΩ, Ei ∧ Ej i. The last constraint is holonomic, and for the physical motion we take rn = ρ. From now on we take SO(n) × Rn−1 for the configuration space of the rolling sphere, where Rn−1 is identified with the affine hyperplane ρΓ + H. The Chaplygin sphere is Rn−1 -Chaplygin system and the reduced space D/Rn−1 is the tangent bundle T SO(n).
ˇ ´ BOZIDAR JOVANOVIC
4
Remark 1. Together to the Chaplygin sphere we can also consider the rubber Chaplygin sphere, defined as a system (4), (6) subjected to the additional right-invariant constraints Ωij = 0, 1 ≤ i < j ≤ n − 1 describing the no-twist condition at the contact point [11, 21]. 2.2. Dynamics. From the constraints (6) we find the form of reaction forces in the right-trivialization in which the equations (2) read
(9)
M˙ = −ρΛ ∧ Γ, ˙ = Λ, mW
(10)
g˙ = Ω · g,
(11)
r˙ = W.
(8)
where M = Adg (Iω) is the ball angular momentum in the space and Λ ∈ Rn is the Lagrange multiplier. ˙ On the other Differentiating the constraints (6) and using (9) we get Λ = mρΩΓ. hand ˙ ∧ Γ = mρ Ω˙ Γ ⊗ Γ + Γ ⊗ Γ Ω ˙ = mρ prh (Ω), ˙ (12) Λ ∧ Γ = mρ(ΩΓ)
where h ⊂ so(n) is the linear subspace
h = Rn ∧ Γ,
(13)
and prh : so(n) → h, prh (ξ) = (ξΓ) ∧ Γ = ξΓ ⊗ Γ + Γ ⊗ Γξ is the orthogonal projection with respect to the scalar product (5). Whence, (8), (10) is a closed system on T SO(n), representing the Chaplygin reduction of Rn−1 symmetry. Now we need to write it in the left trivialization of T SO(n). Let γ = g −1 Γ be the vertical vector in the frame attached to the ball. Then (14)
Adg−1 (h) = Rn ∧ γ =: hγ .
˙ and the relations (12) and pr γ (ξ) = (ξ ·γ)∧γ = From the identity ω˙ = Adg−1 (Ω) h ξ γ ⊗ γ + γ ⊗ γ ξ we get I ω˙ = [Iω, ω] − mρ2 (ω˙ γ ⊗ γ + γ ⊗ γ ω). ˙ Let us denote mρ2 by D and let (15)
k = Iω + D prhγ ω = Iω + D(ω γ ⊗ γ + γ ⊗ γ ω) ∈ so(n)∗
be the angular momentum of the ball relative to the contact point (see [16]). By using the Poisson equation (16)
γ˙ = −ωγ
d it easily follows dt (ωγ ⊗ γ + γ ⊗ γω) = ωγ ˙ ⊗ γ + γ ⊗ γ ω˙ + [ωγ ⊗ γ + γ ⊗ γω, ω]. Therefore, the reduced Chaplygin sphere equations, in variables (ω, g) are given by
(17)
k˙ = [k, ω],
(18)
g˙ = g · ω,
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
5
while the reduced kinetic energy is given by Lred = 12 hk, ωi. The system is additionally left SO(n−1)–invariant where the action of SO(n−1) is given by the rotations around the vertical vector Γ. The closed system (16), (17) in variables (ω, γ) represents the reduction of SO(n − 1)–symmetry to so(n) × S n−1 ∼ = (T SO(n))/SO(n − 1). It possesses an invariant measure with density [16] q (19) µ = det(I + D prhγ ).
We can also consider the system (17), (18) on the cotangent bundle T ∗ SO(n) in ∗ n−1 variables (k, g) or in coordinates (k, γ) on . Then qthe reduced space so(n) × S
the invariant measure is given by µ = 1/ det(I + D prhγ ) (see [13]). In the case n = 3, under the isomorphism between so(3) and R3
(20)
ωij = εijl ωl ,
kij = εijl kl ,
from (17) and (16) we obtain the classical Chaplygin’s ball equations ~k˙ = ~k × ~ω , (21) ~γ˙ = ~γ × ~ω , where ~k = I~ ω + D~ ω − D(~ ω , ~γ )~γ and (22)
I = diag(I1 , I2 , I3 )
is the inertia operator of the ball. In the space (~ω, ~γ ) the density of an invariant measure (19), up to the multiplication by a constant factor is equal to p 1 − D(γ, (I + DI)−1 γ), the expression given by Chaplygin in [8]. 3. Reduced System in Redundant Coordinates
3.1. Reduction to T ∗ S n−1 . From (8) we have d d (23) (prk M ) = (pr Adg (Iω)) dt dt k d d (pr Adg (Iω + D prhγ ω)) = (prk Adg k) = 0, = dt k dt d d (prkγ (Iω)) = dt (prkγ k) = 0, where k ∼ or, equivalently, dt = so(n − 1) is orthogonal complement to h with respect to (5) and kγ := Adg−1 k = (Rn ∧ γ)⊥ . The integral (23) is actually the momentum mapping Φ : T ∗ SO(n) → so(n − 1)∗ ∼ = k,
Φ(k, g) = prk Adg (k)
of the left SO(n − 1)–action. Namely, since for ξ ∈ so(n − 1) the associated vector field ξSO(n)×Rn−1 is a section of (7), we have the Noether conservation low (3). The integral is Rn−1 –invariant and descend to T ∗ SO(n). For n = 3 we have the classical area integral (~k, ~ ω). As in the usual symplectic reduction, we can use the momentum mapping Φ to reduce the system to Mη = Φ−1 (η)/SO(n − 1)η , where SO(n)η is the isotropy
ˇ ´ BOZIDAR JOVANOVIC
6
group of η ∈ so(n − 1)∗ (e.g., see [19]). The reduced space Mη is the Oη bundle over T ∗ S n−1 . Here Oη is the coadjoint orbit of η. We shall consider the simplest but still very interesting case, when we fix the value of the momentum mapping Φ to be zero (24)
prkγ (Iω) = prkγ k = 0.
Whence, both k and Iω belong to (14). Now, let us introduce new variables p, ξ ∈ Rn orthogonal to γ (25)
(γ, p) = (γ, ξ) = 0,
such that k = γ ∧ p,
(26)
ω = I −1 (γ ∧ ξ).
Lemma 1. The variables p and ξ are related via p = ξ − DI −1 (γ ∧ ξ)γ
(27)
Proof. The proof directly follows from the relations (15) and (26). From (27), under the conditions (25), the variable ξ can be uniquely expressed via p and γ. Note that the coordinates (γ, p) can be considered as redundant coordinates of the cotangent bundle of the sphere T ∗ S n−1 realized as a subvariety of R2n defined by constraints φ1 ≡ (γ, γ) = 1,
(28)
φ2 ≡ (γ, p) = 0.
Theorem 2. The reduced Chaplygin–sphere problem on T ∗ S n−1 = Φ−1 (0)/SO(n− 1) is described by the equations (29)
γ˙ = Xγ (γ, p) = −ωγ = −I −1 (γ ∧ ξ(γ, p))γ
(30)
p˙ = Xp (γ, p) = −ωp = −I −1 (γ ∧ ξ(γ, p))p
The equations (29) and (30) preserve the reduced Hamiltonian function 1 1 H(γ, p) = hk, ωi = hγ ∧ p, I −1 (γ ∧ ξ(γ, p))i 2 2 (which is now unique only on the subvariety (28)) and the reduced momentum K(γ, p) = hγ ∧ p, γ ∧ pi = (γ, γ)(p, p) − (γ, p)2 = (p, p).
(31)
Proof of Theorem 2. The equation (29) follows directly from the Poisson equation (16). On the other hand, from the equation (17) we get γ˙ ∧ p + γ ∧ p˙ = [γ ∧ p, ω] =⇒
−ωγpT − p(−ωγ)T + γ ∧ p˙ = γpT ω − pγ T ω − ωγpT + ωpγ T
=⇒
γ ∧ p˙ = ωpγ T + γpT ω = (ωp) ∧ γ
=⇒
p˙ = −ωp + λγ.
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
7
The multiplier λ is equal to zero. Indeed, from (28) we have d φ2 = (γ, ˙ p) + (γ, p) ˙ = (−ωγ, p) + (γ, −ωp) + λ(γ, γ) = λ = 0. dt 3.2. Chaplygin Reducing Multiplier. The reduction of the Chaplygin–sphere problem to T ∗ S n−1 is not a Chaplygin reduction. Nevertheless, the idea of a time reparametrization in order to transform the system to the Hamiltonian form is still applicable. The vector field X = (Xγ , Xp ) of the system (29), (30) can be written in the almost Hamiltonian form iX (w) = dH, where the form w is a non-degenerate 2-form on T ∗ S n−1 , a semi-basic perturbation of the symplectic form ω = (dp1 ∧ dγ1 + · · · + dpn ∧ dγn )|T ∗ S n−1
(32)
(see [19]). Let w be a nondegenerate 2-form on an even dimensional manifold M . For a differential equation x˙ = X that can be written in the almost Hamiltonian form iX w = dH, the Chaplygin multiplier is a nonvanishing function N such that ω ˜ = 1 ˜ N w is closed. Since iX˜ ω ˜ = dH, X = N X, after the time substitution dτ = N dt, d ˜ with respect to the x=X the system x˙ = X becomes the Hamiltonian system dτ symplectic form ω ˜ [24, 7, 25, 11]. More generally, N is the Chaplygin multiplier if there exist a 2-form w0 such that iX w0 = 0 and ω ˜ = N (w − w0 ) is symplectic (see [11]). Then, as above, the system x˙ = X becomes the Hamiltonian system d ˜ ˜. dτ x = X with respect to the symplectic form ω Alternatively, a transparent and classical way to introduce the Chaplygin reducing multiplier for our system is as follows. Let N (γ) be a differentiable nonvanishing positive function in a neighborhood of S n−1 . Consider the coordinate transformation (γ, p) 7−→ (γ, p˜),
p˜ = N p
defined is some neighborhood of T ∗ S n−1 and the symplectic form (33)
ω ˜
= d˜ p1 ∧ dγ1 + · · · + d˜ pn ∧ dγn |T ∗ S n−1 = N (ω + d ln N ∧ dγ1 + . . . d ln N ∧ dγ n )|T ∗ S n−1 .
Then N is a Chaplygin multiplier for the reduced system if the equations (29), (30) in the new time dτ = N (q)dt becomes Hamiltonian with respect to the form ω ˜ . If N is a Chaplygin multiplier then from the Liouville theorem we have (34)
LX˜ (˜ ω n−1 ) = 0
⇐⇒
LX (N n−2 ω n−1 ) = 0,
i.e., the original system has the invariant measure with density N (γ)n−2 .
ˇ ´ BOZIDAR JOVANOVIC
8
3.3. Homogeneous Sphere. In [19], it is proved that the equations of motion of the homogeneous ball on the reduced space Mη = Φ−1 (η)/SO(n−1)η , for any value of the SO(n − 1) momentum mapping, are already hamiltonian. This interesting result, for the motion on T ∗ S n−1 = Φ−1 (0)/SO(n − 1) can be easily derived from Theorem 2. Suppose the inertia operator I is multiplication by a constant ρ > 0. Then the ρ equation (27), under the conditions (25), gives ξ = ρ+D p. The reduced system (29), (30) takes the form γ˙ =
1 p, ρ+D
p˙ = −
(p, p) γ, ρ+D
representing the geodesic flow of the metric of the round sphere multiplied by ρ+D. 4. Hamiltonization In this section we shall perform the Hamiltonization of the reduced Chaplygin sphere (29), (30) for the inertia operator defined on the base Ei ∧ Ej via (35)
I(Ei ∧ Ej ) =
ai aj D Ei ∧ Ej , D − ai aj
1 ≤ i < j ≤ n,
where 0 < ai aj < D, 1 ≤ i, j ≤ n. The form of the inertia operator as well as the form of the Chaplygin multiplier below is motivated by the corresponding formulas in the problem of motion of the n-dimensional Veselova problem as well as the rubber Chaplygin ball given in [17] and [21], respectively. Let A = diag(a1 , . . . , an ). Theorem 3. The reduced Chaplygin sphere equations (29), (30), defined by the inertia tensor (35), read 1 (p, γ) (γ, Ap) (γ, γ) p− A−1 γ + 2 γ− 2 Ap, D D(γ, A−1 γ) D (γ, A−1 γ) D (γ, A−1 γ) (p, A−1 γ) (p, p) (p, Ap) (p, γ) (37) p˙ = p− A−1 γ + 2 γ− 2 Ap. D(γ, A−1 γ) D(γ, A−1 γ) D (γ, A−1 γ) D (γ, A−1 γ)
(36) γ˙ =
Proof. From the definition (35), the angular velocity is given by (38)
ω = I −1 (γ ∧ ξ) = A−1 γ ∧ A−1 ξ −
1 γ ∧ ξ. D
Now, the equation (27), under the conditions (25), can be solved 1 (Ap − (p, Aγ)γ) . D(γ, A−1 γ) Thus ω = A−1 γ ∧ p − γ ∧ Ap/D /D(γ, A−1 γ) and (36), (37) simply follows from (29), (30). (39)
ξ=
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
9
4.1. Invariant Measure. Note that the reduced Hamiltonian 1 1 hγ ∧ p, A−1 γ ∧ p − γ ∧ Api (40) H(γ, p) = −1 2D(γ, A γ) D as well as the system (36), (37) itself, is defined on ˆ 2n = R2n \ {γ = 0}. R
(41)
ˆ 2n the system preserve functions φ1 and φ2 . Also, considered on R Consider the standard spherical coordinates (θ, r) = (θ1 , . . . , θn−1 , r) in Rn (γ) and the corresponding canonical momenta (πθ , πr ) = (π1 , . . . , πn−1 , πr ) in R2n (γ, p) with respect to the canonical symplectic form: dp1 ∧ dγ1 + · · · + dpn ∧ dγn = dπ1 ∧ dθ1 + · · · + dπn−1 ∧ dθn−1 + dπr ∧ dr. Then the volume form in R2n can be represented as (42) Ω = dp1 ∧ dγ1 ∧ · · · ∧ dpn ∧ dγn = (dπ1 ∧ dθ1 ∧ · · · ∧ dπn−1 ∧ dθn−1 ) ∧ dpr ∧ dr, p p where r = (γ, γ) and pr = (γ, p)/ (γ, γ). The coordinates (θ, πθ ) are canonical coordinates (the symplectic form (32) equals dθ1 + · · · + dπn−1 ∧ dθn−1 ) and σ = dπ1 ∧ dθ1 ∧ · · · ∧ dπn−1 ∧ dθn−1 =
1 ω n−1 (n − 1)!
is the canonical volume form on the cotangent bundle T ∗ S n−1 , naturally extended ˆ 2n . to R For example, in the case n = 3, the canonical transformation (γ, p) 7→ (θ, r, πθ , πr ) is given by: γ1
= r cos θ1 ,
γ2
= r sin θ1 cos θ2 ,
γ3
= r sin θ1 sin θ2 ,
p1
= π1 sin θ1 /r + πr cos θ1 ,
p2
= −π1 cos θ1 cos θ2 /r + π2 sin θ2 /(r sin θ1 ) + πr sin θ1 cos θ2
p3
= −π1 cos θ1 sin θ2 − π2 cos θ2 /(r sin θ1 ) + πr sin θ1 sin θ2 .
Proposition 4. The reduced Chaplygin system (36), (37) possesses an invariant measure µ(γ) σ = (A−1 γ, γ)−(n−2)/2 σ .
(43)
ˆ 2n is Proof. The divergence of the vector field X in R n X ∂ γ˙ i ∂ p˙ i (γ, Ap) (γ, A−1 p) (44) div(X) = + Ψ, = (n − 2) + + ∂γi ∂pi D(γ, A−1 γ) D2 (γ, A−1 γ) i=1
where
Ψ=
2(γ, γ) trA−1 trA 2(A−2 γ, γ) + 2 − − ) (γ, p). D(γ, A−1 γ)2 D (γ, A−1 γ)2 D(γ, A−1 γ) D2 (γ, A−1 γ)
ˇ ´ BOZIDAR JOVANOVIC
10
Whence, on the invariant submanifold φ2 = πr = 0, in the view of (36), we get n X ∂ γ˙ i (γ, A−1 γ) ˙ ∂ p˙ i µ˙ = (n − 2) + =− . −1 ∂γi ∂pi (γ, A γ) µ i=1
In other words, the density µ(γ) satisfies the Liouville equation n n X X ∂ q˙i ∂ p˙ i ∂µ =0 +µ + q˙i (45) div(µX) = ∂qi ∂qi ∂pi i=1 i=1
on the manifold φ2 = πr = 0. On the other side, from (42) we obtain
LX (µΩ) = LX (µσ) ∧ dπr ∧ dr + µσ ∧ LX (dπr ∧ dr).
(46)
Since the functions φ1 , φ2 are invariants of the vector field X, the Lie derivatives LX dπr and LX dr equal zero. Further, (45) implies that the left hand side of (46) is also equal to zero on the invariant subvariety φ2 = πr = 0. Thus we conclude LX (µσ)|T ∗ S n−1 = 0 as required. 4.2. Time Reparametrization. According to the constraints (28), instead of (40) we can use the Hamiltonian function 1 (47) H(γ, p) = D(γ, A−1 γ)(p, p) − (p, Ap) 2D2 (γ, A−1 γ)
As follows from Proposition 4 and (34), if the reduced Chaplygin system on T S n−1 is transformable to a Hamiltonian form by a time reparameterization, then p the corresponding reducing multiplier N should be proportional to 1/ (γ, A−1 γ). ∗
Theorem 5. Under the time substitution (48)
dτ = N dt =
D
and an appropriate change of momenta (49)
(γ, p) 7−→ (γ, p˜),
1 p dt (A−1 γ, γ) p˜ =
D
p
1 p (γ, A−1 γ)
the reduced system (36), (37) becomes a Hamiltonian system describing a geodesic flow on S n−1 with the Hamiltonian 1 (50) H(γ, p˜) = D(γ, A−1 γ)(˜ p, p˜) − (˜ p, A˜ p) . 2
Proof. Consider the cotangent bundle T ∗ S n−1 realized as a submanifold of R2n given by (51)
ψ1 ≡ (γ, γ) = 1,
ψ2 ≡ (γ, p˜) = 0.
The canonical Poisson bracket on T ∗ S n−1 with respect to the symplectic form (33) can be described by the use of the Dirac bracket (see [10, 22, 1]): {F, G}d = {F, G} − ({F, ψ1 }{G, ψ2 } − {F, ψ2 }{G, ψ1 })/{ψ1 , ψ2 },
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
11
where {F, G} =
n X ∂F ∂G ∂F ∂G . − ∂γi ∂ p˜i ∂ p˜i ∂γi i=1
ˆ 2n , the bracket {·, ·}d is degenerate and has two Casimir funcConsidered on R tions ψ1 and ψ2 . The symplectic leaf given by (51) is exactly the cotangent bundle T ∗ S n−1 endowed with the canonical symplectic form. Under the mapping (49), the Hamiltonain (47) transforms to (50). With the above notation, the geodesic flow defined by Hamiltonian function (50), in the time τ , is the restriction to (51) of (52)
γi′ =
d γi = {γi , H}d , dτ
p˜′i =
d p˜i = {˜ pi , H}d , dτ
i = 1, . . . , n.
It is convenient to find equations (52) by using the Lagrange multipliers (see [22, 1]). Introduce H ∗ = H − λψ1 − µψ2 . The equations (52) are then given by ∂H ∗ ∂H = − µ˜ p = D(A−1 γ, γ)˜ p − A˜ p − µγ, ∂ p˜ ∂ p˜ ∂H ∗ ∂H p˜′ = − =− + λγ + µ = −D(˜ p, p˜)A−1 γ + λγ + µ˜ p ∂γ ∂γ
γ′ =
where multipliers λ and µ are determined from the condition that the constraint functions ψ1 and ψ2 are integrals of the motion. The straightforward calculations yield (γ, A˜ p) γ, (γ, γ) (˜ p, A˜ p) (γ, A˜ p) p˜′ = −D(˜ p, p˜)A−1 γ + γ− p˜. (γ, γ) (γ, γ) γ ′ = D(A−1 γ, γ)˜ p − A˜ p+
(53) (54)
In the time t, after inverting the mapping (49), the equation (53) takes the form p (γ, Ap) 1 −1 −1 γ , γ˙ · D (γ, A γ) = p D(A γ, γ)p − Ap + (γ, γ) D (γ, A−1 γ)
i.e.,
(55)
γ˙ =
1 1 (γ, Ap) p− 2 Ap + 2 γ, D D (γ, A−1 γ) D (γ, A−1 γ)(γ, γ)
which coincides with (36) on the points of T ∗ S n−1 . Further, ! ! p p p d d d p p p˜ = = D (γ, A−1 γ) = dτ dτ D (γ, A−1 γ) dt D (γ, A−1 γ) ! p 1 (A−1 γ, γ) ˙ 1 d (56) (γ, A−1 γ) = p˙ − p + p˙ p = p p −1 −1 −1 dt (γ, A γ) (γ, A γ) (γ, A γ)
ˇ ´ BOZIDAR JOVANOVIC
12
Finally, subtracting (49) in the left hand side of (54), combining with (55) and (56), we get (57) p˙
=
(p, Ap) (γ, Ap) (p, p) A−1 γ + 2 γ− 2 p −1 −1 D(γ, A γ) D (γ, A γ)(γ, γ) D (γ, A−1 γ)(γ, γ) (γ, Ap) (γ, p) (p, A−1 γ) p+ 2 + p− 2 p −1 −1 2 D(γ, A γ) D (γ, A γ) D (γ, A−1 γ)(γ, γ) −
As above, the equations (37) and (57) are different, but they coincide on the invariant manifold φ1 = ψ1 = 1, φ2 = ψ2 = 0. The theorem is proved. Remark 2. After the isomorphism (20), the operator (35) defines the inertia tensor (22) by I1 = a2 a3 D/(D − a2 a3 ), I2 = a3 a1 D/(D − a3 a1 ), I3 = a2 a3 D/(D − a2 a3 ). In the other direction, given an inertia tensor (22), the matrix A = diag(a1 , a2 , a3 ) is determined via p p i = 1, 2, 3. ai = I1 I2 I3 D(Ii + D)/Ii (I1 + D)(I2 + D)(I3 + D), p The Chaplygin reducing multiplier 1/ 1 − D(γ, (I + DI)−1 γ) given in [6], up to a p multiplication by a constant, coincides with 1/D (A−1 γ, γ). 5. Integrability
5.1. Chaplygin Sphere and the Veselova Problem. The momentum integral (31) after the mapping (49) becomes the integral (58)
K(γ, p˜) = D2 (A−1 γ, γ)(˜ p, p˜)
of the geodesic flow (53), (54). Let 1 (A˜ p, p˜) . 2 Since H = K/2D − G, the quadratic function G is also the integral of the flow. On the other side, (59) is proportional to the Hamiltonian of the reduced multidimensional nonholonomic Veselova problem [17]. More precisely, for the inertia operator defined by
(59)
(60)
G=
−1 I(Ei ∧ Ej ) = a−1 i aj Ei ∧ Ej ,
1 ≤ i < j ≤ n,
the Veselova problem is reducible to T ∗ S n−1 . In redundant coordinates (γ, p) the reduced system is given by (see [17, 18]) 1 1 γ˙ = (−(p, Aγ)γ + (γ, γ)Ap) = (−(p, Aγ)γ + Ap) , (γ, A−1 γ) (γ, A−1 γ) (p, Ap) 1 (−(p, Ap)γ + (p, γ)Ap) = γ. p˙ = − (γ, A−1 γ) (γ, A−1 γ) Furthermore, as it follows from [17, 18], under the time substitution (48) and the change of momenta (49) the reduced system becomes a Hamiltonian system describing a geodesic flow on S n−1 with the Hamiltonian H = D2 G. Let us suppose ai 6= aj , i 6= j. Then the Hamiltonian (59) has the St¨ackel form in sphero-conical variables and the geodesic flow on S n−1 determined by H
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
13
is completely integrable (see [17]). Whence the system (53), (54) is completely integrable as well. Therefore, as in the three-dimensional case [12], we have Theorem 6. The reduced multidimensional nonholonomic Chaplygin–sphere problem defined by inertia operator (35) and the reduced Veselova problem defined by inertia operator (60) have the same invariant toric foliation. Let T be a regular, (n−1)-dimensional invariant torus. Then there exist angle coordinates ϕ1 , . . . , ϕn−1 on T in which both problems simultaneously take the form ϕ˙ 1 = N (ϕ1 , . . . , ϕn−1 ) ω1 , . . . , ϕ˙ n−1 = N (ϕ1 , . . . , ϕn−1 ) ωn−1 with different frequencies ω1 , . . . , ωn−1 . Remark 3. Let x(t) be a geodesic line on the ellipsoid E n−1 = {x = (x1 , . . . , xn ) ∈ Rn | (x, Ax) = 1} endowed with the standard metric. Then, up to the time p rescaling, the unit normal vector γ(t) = Ax(t)/ (Ax(t), Ax(t)) is a solution of the reduced generalized Veselova system defined by inertia tensor (60) (see [18]). In this sense, the momentum integral (58) corresponds to classical Joachimsthal’s integral of the geodesic flow on the ellipsoid E n−1 (see [22]). 5.2. Lagrange Case. The system is integrable even if not all ai are distinct. For any pair of equal parameters ai = aj , the geodesic flow (53), (54) has the additional linear integral fij = γi p˜j − γj p˜i . For example, let n = 4 and a1 = a2 6= a3 = a4 . Then the complete set of commuting integrals is f12 , f34 and H. If we have at least three equal parameters, the system is integrable according to the non-commutative version of the Liouville theorem. We will consider the case a1 = a2 = · · · = an−1 6= an .
(61)
Namely, in general, for n ≥ 4, the operator (35) is not a physical inertia operator of a multidimensional rigid body (see [16]). However, by taking conditions (61) and 2an D > a1 an + a1 D, we get the operator Iω = Jω + ωJ, where J = diag(J1 , J1 , . . . , J1 , Jn ), J1 =
a21 D , 2(D − a21 )
Jn =
a21 D a1 an D − , D − a1 an 2(D − a21 )
representing a SO(n − 1)–symmetric rigid body (multidimensional Lagrange case [2]) with a mass tensor J. Due to the additional SO(n − 1)-symmetry, the geodesic flow (53), (54) has the integrals fij , 1 ≤ i < j ≤ n − 1. Thus, in the original coordinates we get integrals (62)
Fij = (γ, A−1 γ)(γi pj − γj pi )2 ,
1 ≤ i < j ≤ n − 1.
In this case we do not need Hamiltonization to integrate the reduced system, it is already integrable according to the Euler–Jacobi theorem (e.g., see [1]). Since the
14
ˇ ´ BOZIDAR JOVANOVIC
generic invariant manifolds given by H and integrals (62) are two-dimensional and the system has an invariant measure we have [1]: Theorem 7. The Lagrange case of the reduced Chaplygin system (36), (37) is solvable by quadratures. Remark 4. Although the Lagrangian (4) is additionally invariant with respect to the right SO(n − 1)–action, the integrals (62) are not Noether’s integrals. The reason is that the associated vector fields are not sections of the distribution (7). For n = 3 and I1 = I2 , the corresponding integral of the system (21) is F = k32 − D(γ, (I + DI)−1 γ)k32 . Acknowledgments. I am greatly thankful to Yuri N. Fedorov for useful discussions. The research was supported by the Serbian Ministry of Science, Project 144014, Geometry and Topology of Manifolds and Integrable Dynamical Systems. References [1] Arnold V I, Kozlov V V, Neishtadt A I 1985 Mathematical aspects of classical and celestial mechanics. Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamental’nye Napravleniya, Vol. 3, VINITI, Moscow 1985. English transl.: Encyclopadia of Math. Sciences, Vol.3, SpringerVerlag, Berlin 1989. [2] Beljaev A V 1981 Motion of a multidimensional rigid body with a fixed point in a gravitational force field Mat. Sb. 114(156) no. 3, 465-470 (Russian). [3] Bloch A M, Krishnaprasad P S, Marsden J E, Murray R M 1996 Nonholonomical mechanical systems with symmetry Arch. Rational Mech. Anal. 136 21-99. [4] Bloch A M, Zenkov D V 2003 Invariant Measures of Nonholonomic Flows With Internal Degrees of Freedom. Nonlinearity 16, 1793–1807. [5] Borisov A, Mamaev I 2001 Chaplygin’s ball rolling problem is Hamiltonian. (Russian) Mat. Zametki 70, no. 5, 793–795; English translation: Math. Notes 70 (2001), no. 5-6, 720–723 [6] Borisov A, Mamaev I 2007 Isomprphism and Hamiltonizations of Some Nonholonomic Systems, Sib. Mat. Zh. 46, no. 1, 33-45. (see also arXiv:nlin/0509036) [7] Cantrijn F, Cortes J, de Leon M, Martin de Diego D 2002 On the geometry of generalized Chaplygin systems. Math. Proc. Cambridge Philos. Soc. 132 no. 2, 323-351; arXiv: math.DS/0008141. [8] Chaplygin S A 1903 On a rolling sphere on a horizontal plane. Mat. Sbornik 24 139-168 (Russian) [9] Chaplygin S A 1911 On the theory of the motion of nonholonomic systems. Theorem on the reducing multiplier. Mat. Sbornik 28 no. 2, 303-314 (Russian). [10] Dirac P A 1950 On generalized Hamiltonian dynamics. Can. J. Math. 2, no.2, 129–148. [11] Ehlers K, Koiller J, Montgomery R, Rios P 2005 Nonholonomic systems via moving frames: Cartan’s equivalence and Chaplygin Hamiltonization, The breadth of symplectic and Poisson geometry, 75–120, Progr. Math., 232, Birkhuser Boston, Boston, MA, arXiv: math-ph/0408005. [12] Fedorov Yu 1989 Two Integrable Nonholonomic Systems in Classical Dynamics, Vest. Moskov. Univ. Ser I Mat. Mekh. no 4, 38–41 (Russian). [13] Fedorov Yu 1996 Dynamical systems with an invariant measure on the Riemannian symmetric pairs (GL(N ), SO(N )). (Russian) Reg. Ch. Dyn. 1, no. 1, 38–44. [14] Fedorov Yu 2007 A Discretization of the Nonholonomic Chaplygin Sphere Problem, SIGMA 3 (2007), 044, 15 pages, arXiv:nlin/0612037 [15] Fedorov Yu 2009, A Complete Complex Solution of the Nonholonomic Chaplygin Sphere Problem, preprint [16] Fedorov Yu N, Kozlov V V 1995 Various aspects of n-dimensional rigid body dynamics Amer. Math. Soc. Transl . Series 2, 168 141–171.
HAMILTONIZATION OF THE CHAPLYGIN SPHERE
15
[17] Fedorov Yu N, Jovanovi´ c B 2004 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces. J. Non. Sci., 14, 341-381, arXiv: math-ph/0307016. [18] Fedorov Yu N, Jovanovi´ c B 2009 On the Hamiltonization of the Generalized Veselova LR System, preprint [19] Hochgerner S, Garc´ıa-Naranjo L 2008 An (almost) symplectic view of Chaplygin ball, arXiv:08105454 [math-ph] [20] Koiller J 1992 Reduction of some classical non-holonomic systems with symmetry Arch. Rational Mech. 118 113-148. [21] Jovanovi´ c B 2009 LR and L+R systems, arXiv: 0902.1656 [math-ph] [22] Moser J 1980 Geometry of quadrics and spectral theory. Chern Symposium 1979, Berkeley, Springer, 147–188. [23] Schneider D. 2002 Nonholonomic Euler-Poincar´ e Equations and Stability in Chaplygin’s Sphere. Dynamical Systems: An International Journal., 17 No. 2, 87–130 [24] Stanchenko S 1989 Nonholonomic Chaplygin systems. Prikl.Mat.Mekh. 53, no.1, 16–23. English transl.: J.Appl.Math.Mech. 1989 53, no.1, 11–17. [25] Tatarinov Ya 1990 Nonholonomic Systems in Comparatione to Hamiltonian Ones, Doctoral Dissertation, Moscow State University, Moscow. [26] Veselov A P, Veselova L E 1988 Integrable nonholonomic systems on Lie groups Mat. zametki 44 no. 5, 604-619 (Russian); English translation: 1988 Mat. Notes 44 no. 5. Mathematical Institute SANU, Kneza Mihaila 36, 11000, Belgrade, Serbia E-mail address: [email protected]
