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ISSN 1560-3547, Regular and Chaotic Dynamics, 2012, Vol. 17, No. 2, pp. 131­141. c Pleiades Publishing, Ltd., 2012.

On Invariant Manifolds of Nonholonomic Systems
Valery V. Kozlov
*

V.A. Steklov Mathematical Institute Russian Academy of Sciences ul. Gubkina 8, Moscow, 119991 Russia
Received December 27, 2011; accepted January 23, 2012

Abstract--Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely pro jected onto configuration space. The invariance conditions are represented in the form of generalized Lamb's equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered. MSC2010 numbers: 70Hxx, 37J60 DOI: 10.1134/S1560354712020037 Keywords: invariant manifold, Lamb's equation, vortex manifold, Bernoulli's theorem, Helmholtz' theorem

1. INVARIANCE CONDITIONS Consider a nonholonomic system sub ject to potential forces and the constraint (a, x) = b. (1.1) Here, x = (x1 , . . . , xn ) are generalized coordinates, x = (x1 , . . . , xn ) are generalized velocities, a(x, t) is a smooth nonzero covector field and b(x, t) is a scalar function. Let L(x, x, t) = L2 + L1 + L0 be a Lagrangian of the system considered and let Lj be a form homogeneous in velocities of degree j , the quadratic form L2 being positive definite. The equations of motion have the form of the Lagrange equations with Lagrange multiplier L x
·

-

L = µa. x

(1.2)

Equations (1.1)­(1.2) constitute a closed system. Without solving the equations of motion, the multiplier µ can be found as a function of the state of the system x, x and time t. It is useful to pass on to the canonical variables x, y = L , x H + µa, x

where y = (y1 , . . . , yn ) are canonical momenta. Then Eq. (1.2) becomes x=
*

H , y

y=-

(1.3)

E-mail: kozlov@pran.ru

131


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where H (x, y , t) is a Hamiltonian and the multiplier µ is represented as a function of canonical variables x, y and time. The constraint equation (1.1) turns into the linear in momenta integral of the equations of motion (1.3): (A, y ) = B , (1.4) where the vector field A and the scalar function B smoothly depend on x and t. We shall study the invariant n-dimensional manifolds of the equations of motion (1.3), which are uniquely pro jected onto a configuration space {x}: t = {x, y : y1 = u1 (x, t), . . . , yn = un (x, t)}. (1.5) This manifold is given by the momentum field u = (u1 , . . . , un ) on the configuration space. The generalized coordinates x1 , . . . , xn serve as local coordinates on t . We introduce the velocity field v = (v1 , . . . , vn ) with the components vj (x, t) = H yj
y =u(x,t)

and restrict the Hamiltonian H to the manifold (1.5): h(x, t) = H (x, u(x, t), t). It is straightforward to verify [1] that the condition for the manifold (1.5) to be invariant reduces to the equation u h + (rot u)v = - + a, t x where rot u = uj ui - xj xi (1.6)

is a skew-symmetric n â n matrix (vorticity of the covector field u), the multiplier (x, t) is the function µ, in which the canonical momenta are replaced with u(x, t). Ultimately (1.6) is a closed system of n first-order differential equations for finding n functions u1 , . . . , un . We note that the condition for invariance of the manifold (1.5) for the holonomic system in a nonpotential force field also has the form (1.6). For Hamiltonian systems (when a = 0) Eq. (1.6) is called the generalized Lamb equation [2]. The condition (1.6) can be rewritten in invariant notation as: + iv (d ) = -dh + . t Here = uj (x, t) dxj , = aj dxj (1.7)

are differential 1-forms, iv is an inner product of the field v and the form . Applying exterior differentiation to both sides of (1.7), we obtain the vorticity equation + Lv = , (1.8) t where = d and = d are closed differential 2-forms, and Lv = div + iv d is a Lie derivative along the vector field v . Since in the general case = 0, the 2-form is not an invariant of the restriction of the equations of motion (1.3) to the invariant manifold (1.5) x = v (x, t). (1.9) Moreover, the necessary condition for invariance of manifolds of any dimension for the equations of motion of nonholonomic systems can also be represented in the form (1.7) (for more on this see [3], where this result was obtained for Hamiltonian systems).
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2. GENERAL PROPERTIES OF INVARIANCE EQUATIONS As was shown in [2], for Hamiltonian systems the invariance equations (1.6) (or (1.7)) possess many properties characteristic of hydrodynamic equations for an ideal fluid. If = 0, this observation holds only under additional conditions. Consider first the stationary case where b = 0 and the above-mentioned tensor ob jects (u, v , , h, ) do not explicitly depend on time. Obviously, the function h will be constant along the streamlines -- integral curves of the vector field v . Indeed, by virtue of the constraint equation the following relation holds iv = (v ) = µ aj xj = 0.

Applying the operation of interior multiplication iv to both sides of (1.7), we obtain iv dh = h · v = 0, x

which is the required result. As in hydrodynamics, for the Hamilton equations the function h is also constant on the vortex manifolds of the 2-form . In the general case (when = 0) this is not true any more. Recall the definition of the vortex manifold, which is also appropriate for the non-autonomous case. The vortex vector w is determined from the condition iw = 0 . The collection of all vortex vectors at one and the same point x at a fixed time generates some mdimensional vector space x tangential to the configuration space. In a typical case, the dimension m = dim x is independent of the point x. Consequently, there is a distribution of m-dimensional tangential spaces. This distribution is integrable due to the closure property of the 2-form [2]. This means that the configuration space is foliated into m-dimensional manifolds W , which at each point x touch x . These manifolds are called vortex manifolds; in the nonstationary case they depend on time t. Let us continue considering the stationary case. Prop osition 1. If the vortex vectors are virtual displacements, the function h is constant on vortex manifolds. Indeed, since the 2-form = d is skew-symmetric, iw (iv ) = (v , w) = 0. According to the assumption iw = (w) = 0. Hence, h · w, x which means that the function h on the vortex manifolds is constant. In what follows we shall assume the 1-form to have a constant class in the entire configuration space or its part (where an analysis of the system is carried out). This means that the dimension of the linear space of the vectors w, which satisfy the relations 0 = iw dh = iw = iw d = 0, does not change from point to point. It is well known [4, 5] that in some local coordinates the form becomes dS + x1 dx2 + . . . + x Here, S is some function of x and t. Theorem 1. Let us assume that the vortex vectors w of the 2-form satisfy the fol lowing conditions:
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2k-1

dx2k ,

2k

n.

(2.1)


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1. iw = 0 (w is a virtual displacement of the system), 2. iw = 0 (w is a vortex vector of the 2-form ). Then the flow of system (1.9) maps vortex manifolds into vortex manifolds. This statement is a multi-dimensional generalization of the classical Helmholtz theorem on the freezing-in of vortex lines. Proof. Using (2.1), we can write explicitly the equations for invariant manifolds u1 = and Eqs. (1.6) S S + h + 2 , x2 = + h + 1 , x2 t x1 t ............ S S x2k-1 = - + h + 2k , x2k = + h + 2 x2k t x2k-1 t S S + h = 2k+1 , . . . , + h = n . x2k+1 t xn t x1 = - S S , u2 = + x1 , . . . , u2k x1 x2
+1

=

S S , . . . , un = , x2k+1 xn

(2.2)

(2.3)
k-1

, (2.4)

Here, j = aj . According to (2.2), the matrix of the 2-form = d has a block-diagonal form: 0 -1 rot u = (J, . . . , J , 0, . . . , 0), J = . 10 k Hence, the space of vortex vectors is generated by the combinations of n - 2k vectors (0, . . . , 0, 1, 0, . . . , 0)T , . . . , (0, . . . , 0, 0, 0, . . . , 1)T . According to condition 1), 2
k+1

= · · · = n = 0.

(2.5)

But then it follows from (2.4) that the function S +h t and time. By the way, this fact extends the Bernoulli theorem to the

depends only on x1 , . . . , x2k nonstationary case. It is straightforward to verify that the condition 2) of the theorem is equivalent to the series of equations 1 2k+1 n-1 2k+1 = ,..., = , x2k+1 x1 x2k+1 xn-1 ... n n-1 n 1 = ,..., = . xn x1 xn xn-1 Using (2.5), we obtain that the functions 1 , . . . , 2k depend only on x1 , . . . , x2k and t. But then the relations (2.3) will be a closed system of ordinary differential equations in x1 , . . . , x2k . Since the vortex manifolds are locally given by the equations x1 = x0 , . . . , x2k = x0k , 1 2 the flow of the system (1.9) (part of which are Eqs. (2.3)) maps the vortex manifolds into vortex manifolds, which is the required result.
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We emphasize that the system of differential equations (2.3) is closed and has the form (1.6). Its phase space is obtained from configuration space by factorization using a natural equivalence relation: the points lying on one and the same vortex surface are identified. Therefore (2.3) can be called a quotient system. This construction generalizes the definition of the quotient system for invariant manifolds of Hamiltonian systems [2]. As an illustration, we consider a balanced Chaplygin sleigh on a horizontal plane. As generalized coordinates we take the Cartesian coordinates x1 , x2 of the contact point of the blade with ice (in view of the assumption of equilibrium, this point concides with the pro jection of the center of mass of the blade onto the plane of ice) and the rotation angle of the blade. The Lagrangian coincides with the kinetic energy m2 J 2 2 , T= x1 + x2 + 2 2 where m is the mass of the sleigh, J is its moment of inertia relative to the vertical axis passing through the center of mass. The condition of absence of slipping in the horizontal direction orthogonal to the plane of the sleigh is expressed by the nonintegrable constraint equation x1 sin - x2 cos = 0. (2.6) So that the covector a has the components sin , - cos , 0. (2.7) The equations of motion look simple: mx1 = sin , mx2 = - cos , ¨ ¨ This leads us to the formula for the Lagrange multiplier: = -m x2 + x2 . 1 2 ¨ J = 0. (2.8) (2.9)

It is constant throughout the motion of the sleigh. (2.6) and (2.8) imply the presence of a three-dimensional invariant manifold mx1 = cos , mx2 = sin , J = , (2.10) where and are constant parameters whose meaning is obvious. Thus, = cos dx1 + sin dx2 + d . According to (2.7) and (2.9), we have = sin dx1 - cos dx2 , = 0 is some constant. Hence, = d = - sin d dx1 + cos d dx2 . This 2-form is determined by the skew-symmetric 3 â 3 matrix representation of the curl operator 0 - sin 0 0 0 cos . sin - cos 0 Hence, the vector w = (cos , sin , 0)
T

(2.11)

will be a vortex vector. If = 0, all vortex vectors are collinear with the vector (2.11). It is clear that (2.10) will be a virtual displacement of the system: (w) = 0. However, the second condition of the theorem is not satisfied. It can be shown that for = 0 the conclusion of Theorem 1 does not hold: the integral curves of the vector field (2.11) are not frozen into the phase flow of the system (2.10).
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3. SYSTEMS WITH THREE-DIMENSIONAL INVARIANT MANIFOLDS Let n = 3 and assume that the nonholonomic system admits a stationary invariant manifold that is is uniquely pro jected onto configuration space. Eqs. (1.1) and (1.6) then become h + a , (a , v ) = 0. (3.1) x Here, for brevity of notation, a denotes the product a. Further, we consider the most interesting case where the vector field v does not vanish. Then the second equation of (3.1) implies that the covector field a can be expressed as the product â v , where is some field on configuration space. The representation (rot u) â v = - a = â v, (3.2) is, of course, not single-valued: one can add the term Ôv to the field , where Ô(x) is any smooth function. Prop osition 2. Assume that the fol lowing conditions are satisfied: 1) the autonomous system (1.9) admits an invariant measure with a smooth density > 0: div(v ) = 0, 2) div = 1 2 3 + + = 0. x1 x2 x3 rot u -

Then the vector fields v commute (their Lie bracket is zero). Indeed, using (3.2), the first equation of (3.1) becomes ( - rot u) â v = with div( - rot u) = 0. As was established in [6], by virtue of the first condition the fields (3.4) commute. The vectors (3.3) may turn out to be collinear at each point of the configuration space. According to (3.4), this is obviously the case if the function h assumes a constant value. For example, in the problem of a balanced sleigh (from Section 2) h = const on the invariant manifold (2.10). The function h is a first integral of the autonomous system on the configuration space x = v (x). Consider the "Bernoulli surfaces" Bc = {x : h(x) = c}. (3.6) If c does not coincide with the critical value of the function h, then Bc will be a smooth regular surface. If v = 0 on Bc , each connected compact component of the Bernoulli surface will be a two-dimensional torus. What is the structure of the phase flow of system (3.5) on the integral surface (3.6)? Theorem 2. Assume that the conditions of Proposition 2 are satisfied and c is not a critical point of the function h. Then on each connected compact component of the Bernoul li surface one can introduce angular coordinates 1 , 2 mod 2 such that the vector field (3.5) restricted to the manifold (3.6) simplifies to 1 = 1 , 2 = 2 ; j = const. Thus, on the compact regular Bernoul li surfaces the system executes conditional ly periodic motions.
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and

(3.3)

h , x

(3.4)

(3.5)


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Proof of Theorem 2. By (3.4), the fields (3.3) touch the surface (3.6) at all points and are linearly independent, since dh = 0 on Bc . On the other hand, these fields commute on Bc (Proposition 2). This implies the existence of angular variables uniformly changing with time (see, e.g., [7]). The conditions of Proposition 2 are essential to the validity of the conclusion of Theorem 2. Without these additional conditions the structure of the phase flow of system (3.5) on the invariant tori (3.6) can be fairly complicated. It should also be noted that the first condition of Proposition 2 is seldom satisfied, since in a typical case the equations for nonholonomic dynamical systems do not at all admit any integral invariants with smooth density [8]. 4. SYSTEMS ON LIE GROUPS WITH LEFT-INVARIANT CONSTRAINTS Let v1 , . . . , vn (4.1) be vector fields on a configuration manifold M n , which are linearly independent at all points. Their commutators can be written as [vi , vj ] = ck (x)vk , ij
n

ck = -cki . ij j

Let us introduce the quasi-velocities = (1 , . . . , n ) using the formula x=
i=1

vi (x)i .

(4.2)

The Lagrangian L and the constraint equation f (x, x) = 0 can be written in terms of the variables x, ; let these be the functions L( , x) and f ( , x). The Lagrange equation then becomes L i
· n

=
j,k=1

cki j j

L f + vi (L) + . k i

(4.3)

Here, vi (L) denotes the derivative of the Lagrangian along the vector field vi . Eq. (4.3) must be supplemented with the constraint equation f ( , x) = 0. For constraints linear in velocities: f ( , x) = (, ), where the field smoothly depends on x. Equations (4.3) (in the absence of constraints) were derived by Poincar´ All this can be easily extended to the case where the Lagrangian of the system e. and the constraint equation depend on time. Now let M be a Lie group G, and let (4.1) be independent left-invariant fields on G. Then the coefficients ck are the structure constants of the Lie algebra g of the group G. Assume that the ij Lagrangian is invariant under left translations on the group. In particular, vi (L) = 0 (1 i n) and the Lagrangian L depends on the quasi-velocities (vectors from the algebra g ). Moreover, if the constraint is left-invariant, the function f does not depend on the coordinates x1 , . . . , xn either. In this case, the system of equations (4.3)­(4.4) becomes a closed system of differential equations on the algebra g : L i
·

(4.4)

=

cki j j

L f + , k i

f ( ) = 0.

(4.5)

In [9], where nonholonomic systems on Lie groups were studied, Eqs. (4.5) were called the Euler­ e­Suslov equations. G. K. Suslov himself derived these equations in the case where G Poincar´ coincides with the rotation group S O(3) in three-dimensional Euclidean space. He considered the
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problem of rotation of a rigid body with a non-integrable constraint: the pro jection of angular velocity onto some body-fixed direction is zero [10]. Introducing the quasi-momenta m = (m1 , . . . , mn ) g , mi = L , i (4.6)

Equations (4.5) can be recast as a closed system of equations on the dual algebra g :
n

mi =
j,k=1

cki mk i + j

f = 0,

f ( ) = 0.

(4.7)

The quasi-velocities are expressed in terms of quasi-momenta by the well-known formulae: k = H , mk H = (m · - L)
m

.

As already mentioned, the system (4.5) (accordingly, (4.7)) is closed with respect to the quasivelocities (quasi-momenta): the multiplier can be found as a function of (or of m). This simple observation allows us to establish a whole family of invariant n-dimensional manifolds of nonholonomic systems with left-invariant constraints. Indeed, let t (t) be one of the solutions to the system (4.5). Then the invariant manifold can be identified with the configuration manifold, Eqs. (1.9) coincide with Eqs. (4.2), and the function h depends only on time and, therefore, does not appear in the invariance equation (1.6). We note an important property of the system (4.2): its flow preserves the right-invariant measure on the group G [2]. Recall that on each Lie group there is a unique (up to a constant multiplier) measure which is invariant under all left (right) translations. In the case of a unimodular group this measure (referred to as the Haar measure) is biinvariant. The analytic criterion for unimodularity requires that the following relations for the structure constants of the Lie algebra be satisfied: ck = 0, ik 1 i n.

All compact Lie groups are unimodular. As an illustration, consider the classical Suslov problem of rotation of a rigid body with a fixed point. Let the pro jection of the body's angular velocity onto the third axis of a movable trihedron be equal to zero: 3 = 0. We do not assume that this trihedron coincides with the principal axes of inertia of the rotating body. It is well known that the remaining two components 1 and 2 are expressed in terms of the elementary time functions and as t + they tend to the constant 0 0 quantities 1 and 2 [10]. Let I = Iij be the inertia tensor of a rigid body relative to the chosen movable trihedron. As generalized variables we choose the ordinary Euler angles , , . The Lagrangian coincides with the kinetic energy 1 Iij i j , T= (4.8) 2 and the components of the angular velocity 1 and 2 are expressed in terms of the Euler angles and their derivatives using the kinematic Euler formulae. The conjugate canonical momenta are introduced according to the ordinary rule: p = T , p = T , p = T .

Using (4.8) and the kinematic Euler formulae, we obtain: p = m1 cos - m2 sin , p = m3 , p = m1 sin sin + m2 sin sin + m3 cos ,
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(4.9)

2012


ON INVARIANT MANIFOLDS OF NONHOLONOMIC SYSTEMS

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where m1 , m2 , m3 the formulae (4.9) The vector field and then imposing

are the pro jections of the body's angular momentum onto the movable axes. In these momenta are taken to be known functions of time. v on the group S O(3) can be defined by inverting the kinematic Euler formulae the constraint 3 = 0)

= 1 cos - 2 sin , 1 sin + 2 cos =- cos ,. (4.10) sin 1 sin + 2 cos . = sin It can be seen that div(v ) = 0, where = sin . Therefore, the flow of the non-autonomous system (4.10) preserves the Haar measure sin d d d (4.11) on the rotation group. Taking the curl of the covector field (4.9) using the standard rule, we find the vortex vector: w = (m1 sin cos - m2 sin sin , -m1 cos sin - m2 cos cos + m3 sin , m1 sin + m3 cos ). (4.12) The components of w appear unwieldy and an interpretation of this vector field on the rotation group S O(3) is not immediately evident. However, this field can be represented in a movable space, using again the kinematic Euler formulae. The pro jections of the vector field (4.12) onto the movable axes are 1 = m1 sin , 2 = m2 sin , 3 = m3 sin . (4.13) Since the vortex vectors are defined up to a nonzero multiplier, the sine of the nutation angle (density of the Haar measure (4.11)) in the formulae (4.13) can be dropped. But then at each moment of time the components i will not depend on the orientation of the rotating rigid body. The field (4.13) determines the rotation of the body about the axis (in the body and in the fixed space), which is collinear with the vector (1 , 2 , 3 ). The corresponding integral curves of the field (4.12) are vortex lines -- large circles on the group S O(3). The foliation of the group S O(3) into these closed vortex lines is well known in topology: this is Seifert's foliation. Factorization by these circles yields a two-dimensional sphere. It should be emphasized that since the field of the curl (4.12), divided into sin , is left-invariant, the vortex lines are invariant under right translations on the group S O(3). The family of right translations includes, among other things, transformations from the flow of the system (4.10). 5. SYSTEMS ON LIE GROUPS WITH RIGHT-INVARIANT CONSTRAINTS The equations of motion for a system on the Lie group G with left-invariant kinetic energy and a right-invariant stationary constraint have the form (4.7):
n

mi =
j,k=1

cki mk j + i , j

(, ) = 0.

(5.1)

But now the covector field on the Lie group will be mapped onto itself under all right translations. Such systems are introduced in [11]. In our case, the following key property holds. Let w be a right-invariant vector field on G. Hence, the phase flow of the system x = w(x), xG will be a family of left translations on G. But, according to the assumption, the kinetic energy is invariant under all left translations. Therefore, if we additionally require the equality (, w) = 0,
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(5.2)


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KOZLOV

then, by the generalized Noether theorem [12], Eqs. (5.1) admit a first integral linear in the velocities (m, w) = const. (5.3)

This circumstance will allow us to show a family of stationary invariant manifolds for Eqs. (5.1), which are uniquely pro jected onto the Lie group G. Indeed, there are n - 1 linearly independent right-invariant vector fields satisfying the condition (5.2). They generate n - 1 linear integrals of the form (5.3). Adding to them the constraint equation, we obtain a nondegenerate linear system of equations in n variables m1 , . . . , mn . Thus, the angular momentum becomes a function on the group which also depends on n - 1 parameters. Of course, this observation also holds for systems with any number of right-invariant constraints. The study of flows on these manifolds is a non-trivial task. Prop osition 3. The flow on the above-mentioned n-dimensional invariant manifold admits an invariant measure with smooth positive density. As an example, consider the problem of rotation of a top with the following nonintegrable constraint: the pro jection of the angular velocity onto some fixed axis vanishes [12]. This constraint is invariant under all right translations on the group S O(3). Let , , be an orthonormal basis in fixed space, while ( , ) = 0 (5.4) is interpreted as a constraint equation. The equations of rotation of a top in a moving reference frame are I + â I = , ( , ) = 0, (5.5) + â = 0, + â = 0, + â = 0. Here, I is the inertia operator of a rigid body relative to a fixed point. Equations (5.5) admit two Noether integrals (I , ) = c1 , (I , ) = c2 . constitute a closed linear system, which in terms of its position. Thus, we obtain a uniquely pro jected onto the group S O(3). Then
-1

Along with the constraint equation, these two relations allows us to express the angular velocity of a rotating top three-dimensional stationary invariant manifold, which is Without loss of generality, we may set c1 = c, c2 = 0. =c I
-1

-

(I (I

-1 -1

, ) I , )

.

(5.6)

The tangent vector field v on the group S O(3) is generated by the kinematic Poisson equations (5.5), into which the substitution (5.6) should be made. The corresponding dynamical system on S O(3) admits an invariant measure with smooth density, as do the initial equations of motion. Moreover, there is the first integral h , which restricts the kinetic energy to the invariant manifold. If c = 0, then h is a nonconstant function on the group S O(3). The corresponding Bernoulli surfaces will again be two-dimensional tori, but the flow on them cannot be reduced to a conditionally periodic form. REFERENCES
1. Arzhanykh, I. S., Momentum Fields, Tashkent: Nauka, 1965 (Russian). 2. Kozlov, V. V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin­Heidelberg: Springer, 2003. 3. Kozlov, V. V., On Invariant Manifolds of Hamilton's Equations, J. Appl. Math. Mech., 2012, in press. (Russian). ´ 4. Cartan, E. J., Le¸ cons sur les invariants int´ aux, Paris: Hermann et fils, 1922. egr 5. Godbillon, C., G´ ´ eometrie diff´ entiel le et m´ er echanique analytique, Paris: Hermann, 1969. 6. Kozlov, V. V., Notes on Steady Vortex Motions of Continuous Medium, Prikl. Mat. Mekh., 1983, vol. 47, no. 2, pp. 341­342 [J. Appl. Math. Mech., 1983, vol. 47, no. 2, pp. 288­289].
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7. Arnold, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1­291. 8. Kozlov, V. V., On the Existence of an Integral Invariant of a Smooth Dynamic System, Prikl. Mat. Mekh., 1987, vol. 51, no. 4, pp. 538­545 [J. Appl. Math. Mech., 1987, vol. 51, no. 4, pp. 420­426]. 9. Fedorov, Yu. N. and Kozlov, V. V., Various Aspects of N -dimensional Rigid Body Dynamics, Dynamical Systems in Classical Mechanics, Amer. Math. Soc. Transl. Ser. 2, vol. 168, Providence, RI: AMS, 1995, pp. 141­171. 10. Suslov, G. K., Theoretical Mechanics, Moscow­Leningrad: Gostekhizdat, 1951 (Russian). 11. Veselov, A. P. and Veselova, L. E., Flows on Lie Groups with a Nonholonomic Constraint and Integrable Non-Hamiltonian Systems, Funktsional. Anal. i Prilozhen., 1986, vol. 20, no. 4, pp. 65­66 [Funct. Anal. Appl., 1986, vol. 20, no. 5, pp. 308­309]. 12. Kozlov, V. V. and Kolesnikov, N. N., On Theorems of Dynamics, Prikl. Mat. Mekh., 1978, vol. 42, no. 1, pp. 28­33 [J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 26­31].

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