LR and L+R Systems

LR and L+R Systems

Božidar Jovanović
Abstract

We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study -dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrization becomes an integrable Hamiltonian system on the –dimensional sphere. Also, we showed that an arbitrary L+R system introduced by Fedorov in [15] can be seen as a reduced system of an appropriate coupled LR system.111AMS Subject Classification 37J60, 37J35, 70H45

1 Introduction

In this paper we study nonholonomic geodesic flows on direct product of Lie groups with specially chosen right-invariant constraints and left-invariant metrics.

Let be a –dimensional Riemannian manifold with a nondegenerate metric and let be a nonintegrable –dimensional distribution on the tangent bundle . A smooth path is called admissible (or allowed by constraints) if the velocity belongs to for all . Let be some local coordinates on in which the constraints are written in the form

(1.1)

where are independent 1-forms. The admissible path is called nonholonomic geodesic if it is satisfies the Lagrange–d’Alambert equations

(1.2)

where the Lagrange multipliers are chosen such that the solutions satisfy constraints (1.1) and the Lagrangian is given by the kinetic energy . After the Legendre transformation , , one can also write the Lagrange-d’Alambert equations as a first-order system on the cotangent bundle . As for the Hamiltonian systems, the Lagrangian (or the Hamiltonian in the cotangent representation of the flow) is always the first integral of the system.

Suppose that a Lie group acts by isometries on (the Lagrangian is - invariant) and let be the vector field on associated to the action of one-parameter subgroup , . The following version of the Noether theorem holds (see [1, 2]): if is a section of the distribution then

(1.3)

On the other side, let be transversal to , for all . In addition, suppose that has a principal bundle structure and that is the collection of horizontal spaces of a principal connection. Then the nonholonomic geodesic flow defined by is called a -Chaplygin system. The system (1.2) is -invariant and reduces to the tangent bundle (for the details see [26, 2, 8, 11]).

The equations (1.2) are not Hamiltonian. However, in some cases they have a rather strong property – an invariant measure (e.g, see [1, 27, 4]). Within the class of -Chaplygin systems, the existence of an invariant measure is closely related with their reduction to a Hamiltonian form after an appropriate time rescaling (see [10, 29, 18, 8, 11]).

Veselov and Veselova [30, 31] constructed nonholonomic systems on unimodular Lie groups with right-invariant nonintegrable constraints and left-invariant metrics, so called LR systems, and showed that they always possess an invariant measure. Similar integrable nonholonomic problems on Lie groups, with left and right invariant constraints, are studied in [17, 21, 22, 3, 19]. Recently, a nontrivial example of a nonholonomic LR system, which can be regarded also as a generalized Chaplygin system (-dimensional Veselova rigid body problem [30, 17]) such that Chaplygin reducibility theorem is applicable for any dimension is given by Fedorov and Jovanović [18].

It appears that LR systems can be viewed as a limit case of certain artificial systems (L+R systems) on the same group, which also possess an invariant measure (see Fedorov [15]). The latter systems do not have a straightforward mechanical or geometric interpretation and arise as a “distortion” of a geodesic flow on whose kinetic energy is given by a sum of a left- and right-invariant metrics.

A class of L+R systems on can be seen as a reduction of a class of nonholonomic systems defined on the semi-direct product of the group and a vector space (see Theorems 3, 4 in Schneider [28]). We shall prove that an arbitrary L+R system on can be obtained as a reduction of a coupled noholonomic LR system defined on the direct product .

One of the best known examples of integrable nonholonomic systems with an invariant measure is the celebrated Chaplygin sphere which describes a dynamically non-symmetric ball rolling without sliding on a horizontal plane and the center of the mass is assumed to be at the geometric center [9]. It is interesting that the Chaplygin’s sphere appears within both constructions. In the construction described in [28] one should take for the configuration space the Lie group of Euclidean motion , that is the semi-direct product of and [28]. On the other side, Chaplygin sphere is a LR system on the direct product (e.g., see [16]). This was a starting point in considering the coupled nonholonomic LR systems below.

Outline and results of the paper.

In Section 2 we recall the definition and basic properties of LR and L+R systems. We define the coupled LR systems and show that any L+R system can be obtain as a reduction of an appropriate coupled LR system (Sections 3, 4). An example of a coupled LR system on is given, which provides an alternative generalization of the Chaplygin sphere problem (Section 4, system (6.17) in Section 6).

In Section 5 we study a -dimensional variant of the spherical support system introduced by Fedorov [13]: the motion of a dynamically nonsymmetric ball with the unit radius around its fixed center that touches arbitrary dynamically symmetric balls whose centers are also fixed, and there is no sliding at the contacts points.

Recall that the rubber rolling of the sphere over some other fixed convex surface in means that that in the addition to the constraint given by the condition that the velocity of the contact point is equal to zero, we have no-twist condition that rotations about the normal to the surface are forbidden. The rubber rolling of the dynamically non-symmetric sphere over the another sphere, considered as a Chaplygin system on the bundle (where acts diagonally on the total space), as well as the Hamiltonization in sphero-conical variables of is given by Koiller and Ehlers [12]. The integrable cases are found by Borisov and Mamaev [7]. In particular, when the radius of the fixed sphere tends to infinity, we get the rubber rolling of the sphere over the plane (rubber Chaplygin sphere). The Chaplygin reducing multiplier for the rubber Chaplygin sphere is given in [11].

By the analogue, we define the -dimensional rubber spherical support system with additional no-twist conditions at the contact points. It appears that both systems fits into the construction of coupled LR systems. Similarly as for the 3-dimensional spherical support system studied in [13], we prove that the 3-dimensional rubber spherical support system is integrable (Section 5).

Finally in Section 6 we consider the -dimensional rubber Chaplygin sphere problem describing the rolling without slipping and twisting of an -dimensional ball on an -dimensional hyperplane in as coupled LR systems on the direct product . It appears that the rubber Chaplygin sphere is a -Chaplygin system closely related to the -dimensional nonholonomic Veselova problem, which allows as to prove the existence of the Chaplygin multiplier for a specially chosen inertia operator of the ball. In particular, when , the multiplier exist for any inertia tensor of the ball, and reduces to the one obtained in [11, 12].

2 Preliminaries

LR systems.

LR system on a Lie group is a nonholonomic geodesic flow of a left-invariant metric and right-invariant nonintegrable distribution (see [30, 31]). Through the paper we suppose that all considered Lie groups have bi-invariant Riemannian metrics, or equivalently -invariant Euclidean scalar products on corresponding Lie algebras. In particular, Lie groups are unimodular.

Let be the Lie algebra of . In what follows we shall identify and by means invariant scalar product , and and by the bi-invariant metric. For clearness, we shall use the symbol for the elements in and the symbol for the elements in .

The Lagrangian is defined by where is the angular velocity in the moving frame. Here is a symmetric positive definite (with respect to ) operator. The corresponding left-invariant metric will be denoted by . The distribution is determined by its restriction to the Lie algebra and it is nonintegrable if and only if is not a subalgebra of . Let be the orthogonal complement of with respect to and let be a orthonormal base of . Then the right-invariant constraints can be written as

or, equivalently,

(2.1)

Here represents angular velocity in the space.

Equations (1.2) in the left trivialization take the form

(2.2)
(2.3)

where is the angular momentum in the body frame.

The Lagrange multipliers can be found by differentiating the constraints (2.1). They are actually defined on the whole phase space and we can consider the system (2.2), (2.3) on as well (see [31]). The constraint functions are then integrals of the extended system and the nonholonomic geodesic flow is just the restriction of (2.2), (2.3) onto the invariant submanifold (2.1).

Instead of (2.2), (2.3), one can consider the closed system consisting of (2.2) and

(2.4)

on the direct product . Let , where is the orthogonal projection to . Then the system (2.2), (2.4) has an invariant measure with density (see [31]).

Also, since for , the associate vector field of the left -action is right invariant and the momentum mapping of the left action equals to (angular momentum in the space), the LR system (2.2), (2.3) has the Noether conservation laws:

(2.5)

If the linear subspace is the Lie algebra of a subgroup , then the Lagrangian and the right-invariant distribution are invariant with respect to the left -action. As a result, the LR system can naturally be regarded as a -Chaplygin system [18].

Geodesic flow on with L+R metric.

In addition to the nondegenerate linear operator defining the left-invariant metric , introduce a constant symmetric linear operator defining a right-invariant metric on the -dimensional compact Lie group : for any vectors we put . We take the sum of both metrics and consider the corresponding geodesic flow on described by the Lagrangian

where . We can also consider the case when is not positive definite, but the total inertia operator is nondegenerate and positive definite on the whole group .

The geodesic motion on the group is described by the Euler–Poincaré equations

(2.6)

together with the kinematic equation .

In order to find explicit expression for , we first note that for any , where is the left-invariant vector field on generated by . Since the metric is left-invariant, we have

As a result, .

Also, in view of the definition of , its evolution is given by matrix equation

(2.7)

Since is invariant scalar product, we have  , and .

Equations (2.6), (2.7) form a closed system on the space with the coordinates (, ), where is a orthonormal base of .

L+R systems.

Following Fedorov [15], consider the equations (2.6) modified by rejecting the term . As a result, we obtain the another system

(2.8)

on , or the system

(2.9)

on the space . This is generally not a Lagrangian system, and, in contrast to equations (2.6), (2.7), it possesses the “momentum” integral  . In view of the structure of the kinetic energy, we shall refer to the system (2.8) (or (2.9)) as L+R system on [15].

The L+R system (2.9) possesses also the kinetic energy integral and an invariant measure (in coordinates , ) with density (see [14, 15]).

As mentioned above, a nonholonomic LR system on a Lie group can be obtained as a limit case of a certain L+R system on this group. Indeed, suppose that the operator defining a right-invariant metric on is degenerate and has the form , , , where, as in (2.1), are orthonormal right-invariant vector fields , . The L+R system (2.9) on the space can be represented in form

(2.10)

Then the following statement holds (see [15]). As , the equations (2.10) transform to the equations with multipliers (2.2) and constraints (2.1), where .

3 Coupled nonholonomic LR Systems

Define a coupled nonholonomic LR system on the direct product () as a LR system given by the Lagrangian function

(3.1)

and right-invariant constraints

(3.2)
(3.3)

where , are mutually orthogonal linear subspaces of .

Here , is the angular velocity in the body and is the angular velocity in the space, . The constant is greater than zero, while , are arbitrary non-zero, real parameters.

The Lagrangian (3.1) in the second variable is right-invariant as well. It is convenient to write the equations of motion both in the left-trivialization (in variables and ) and right-trivialization (in variables and )

(3.4)

Then the right-invariant distribution is given by

Let and let be the orthogonal projections, .

Proposition 3.1.

The admissible path is a motion of the nonholonomic LR system (3.1), (3.2), (3.3) if it satisfies equations

(3.5)
(3.6)
(3.7)
(3.8)

where and

Proof. The equations of a motion in the right-trivialization (or in the space frame) read

(3.9)
(3.10)
(3.11)
(3.12)

where the Lagrange multipliers (reaction forces) belong to () and is the first component of angular momentum in the space frame (the second component is ).

Differentiating the constraints (3.3), from (3.10) we obtain

that is

(3.13)

The equation (3.6) follows from (3.10), (3.13) and the relation

(3.14)

From (3.13) and identities (3.14), and

the equation (3.9) in the left-trivialization takes the form

(3.15)

Now it remains to find the Lagrange multiplier . Differentiating (3.2) we get

Whence, according (3.15) it follows . The proof is complete.

The Lagrangian (3.1) as well as constraints (3.3) are right ()-invariant and the equations (3.5), (3.6), (3.7) can be seen as a reduction of the system to

Let be the right-invariant distribution defined by (3.2).

Theorem 3.2.

The equations (3.5), (3.6), (3.7) on are reducing to the following system on :

(3.16)

Proof. The equations (3.5) and (3.7) form a closed system on . If is a solution of (3.5), (3.7), then one can easily reconstruct the motion of . Let

(3.17)

From (3.7) we have

(3.18)

while the -components of the angular velocity are determined from the constraints (3.3):

Now, let be the orthonormal base of . Then will be the orthonormal base of . We have

Whence, by using (2.4) and the identity , we obtain

The above equation implies that (3.5), (3.7) can be rewritten in the form (3.16).

The derivation of along the flow is: The first term is equal to zero since is a -invariant scalar product, while the second term is equal to zero from the constraint (3.2). We can refer to as to the reduced Lagrangian, or reduced kinetic energy. If , the reduced kinetic energy coincides with the kinetic energy of the reconstructed motion on the whole phase space.

From the equation (3.9) we also get the linear conservation law

(3.19)

The integrals (3.18) and (3.19) are actually Noether integrals (2.5) of the system. The other Noethers integrals are trivial:

Remark 3.1.

If , i.e., we do not impose the constraint (3.2), the reduced system is an L+R system on the Lie group

(3.20)

Further suppose that (3.17) is the Lie algebra of the closed Lie subgroup and that linear subspaces are -invariant:

Then, since , , the L+R equations (3.20) are left -invariant and we can reduce them to , where is the homogeneous space, with respect to the left-action of .

Remark 3.2.

In the case when is the Lie algebra of a closed subgroup , and linear spaces are invariant, then the coupled LR system (3.1), (3.2), (3.3) is -Chaplygin system with respect to the action:

The reduced space is the tangent bundle of the homogeneous space .

Theorem 3.3.

An arbitrary L+R system (2.8) can be seen as a reduction of an appropriate coupled LR system.

Proof. Let be the orthonormal base of in which the symmetric operator has the diagonal form: Then the right invariant term in (2.8) reads , where are given by

(3.21)

Consider the -coupled nonholonomic LR system (3.1), (3.3), where and are the lines in the directions of , . We can choose parameters , , such that , . The system represents a - Chaplygin system with reduced equations of the required form (2.8).

4 -Coupled Systems

There is a straightforward generalization of the construction to the case when we have coupling with different Lie groups, that is the configuration space is the direct product and the Lagrangian is

(4.1)

where are invariant scalar products on Lie algebras , .

Let us fix a base of and some bases of (). Let

be the linear mappings with matrixes () and () in the above bases. In addition, we suppose that the ()-matrixes

are invertible. Consider the right invariant constraints given by

(4.2)

Here, , and , are velocities in the left and right trivializations, respectively and , are real parameters.

Let , denote the column matrix, representing and in the chosen bases. We have , where is the column, representing in the base (3.21).

In the right-trivialization, the equation in reads

(4.3)

where is the Lagrange multiplier ()-matrix. Differentiating the constraints (4.2), from (4.3) we get

(4.4)

Repeating the arguments of Theorems 3.1 and 3.2, the considered -coupled nonholonomic system reduces to the L+R system

where , and in the matrix form, relative to the base (3.21), is given by

As above, one can easily incorporate an additional right invariant constraint of the form (3.2).

LR systems on .

As an example, consider the case where are all equal to the Lie algebra considered as a Abelian group, and the constraints (4.2) are given by

(4.5)

where are fixed elements of the Lie algebra and are real parameters. Note that, since is Abelian group, the angular velocities coincide with the usual velocity: , .

The equations of a motion in the right-trivialization read

(4.6)
(4.7)

where . This is a –Chaplygin system and it is reducible to . Differentiating the constraints (4.5), from (4.7) we get the Lagrange multipliers

Therefore, the equations (4.7) in the left-trivialization take the form

where , . Next, from the identities

we obtain the following proposition

Proposition 4.1.

The reduced equations of the –coupled nonholonomic system (4.1), (4.5) are given by the L+R system

(4.8)

where

Remark 4.1.

Nonholonomic systems on semi-direct products , where is a representation of the Lie group on the vector space are studied in Schneider [28]. Proposition 4.1 can be derived from Theorem 3 given in [28].

5 Spherical Support

Consider the motion of a dynamically nonsymmetric ball in with the unit radius around its fixed center. Suppose that the ball touches arbitrary dynamically symmetric balls whose centers are also fixed, and there is no sliding at the contacts points. We call this mechanical construction the spherical support. For spherical support is defined by Fedorov [13, 15].

The configuration space is : the matrixes map the frames attached to the ball and the th th peripheral ball to the fixed frame, respectively. The Lagrangian is of the form (4.1), where for we take the scalar product proportional to the Killing form

(5.1)

the angular velocities , , , of the balls are defined as above, is the inertia tensor of the ball and are the central inertia moment and the radius of the th peripheral ball.

Let be the unit vector fixed in the space and directed from the center of the ball to the point of contact with the th ball. Nonholonomic constraints express the absence of sliding at the contact points. This means that velocity of the point of contact of the ball with the th ball, in the space frame, is the same as the velocity of the corresponding point on the th ball.

Figure 5.1: The spherical support

Consider the fixed point on the ball with coordinates and in the body and space frames, respectively. Then the velocity of the point in space is given by the Poisson equation (e.g, see [17]) Therefore, the velocity of the contact point with the th peripheral ball is given by . Similarly, the velocity of the corresponding contact point of the th ball in the space frame is given by and the constraints are

(5.2)

We see that the -dimensional spherical support is actually a -coupled LR system studied in the previous section. Let

(5.3)

be the contact points of with the th ball () in the frame attached to the ball . Then the right-invariant constraints (5.2) can be rewritten in the form

(5.4)

where

are linear (no mutually orthogonal) subspaces of the Lie algebra .

From the identity the equations of the motion become

We have the conservation laws

which together with the right -symmetry lead to the following statement

Proposition 5.1.

The spherical support system reduces to the L+R flow

(5.5)

where and are defined by (5.3).

One can say that the reduced system (5.5) on describes the free rotation of a “generalized Euler top”, whose tensor of inertia is a sum of two components: one is fixed in the body and the other one is fixed in the space.

Note that the vectors in the frame attached to the ball satisfy the Poisson equations (e.g., see [17])

(5.6)

By introducing , from (5.6) we obtain

(5.7)

Combining (5.5) and (5.7) we get family of integrals - the coefficients of the polynomials

(5.8)

For the system is integrable by the Euler–Jacobi theorem, and its generic invariant manifolds are two-dimensional tori (see [13, 15]).

Remark 5.1.

If the positions of peripheral balls are mutually orthogonal

then the components of can be seen as redundant coordinates on the Stiefel variety . The system is invariant with respect to the action, representing the rotations in the space orthogonal to