Geometrisation of Chaplygin’s reducing multiplier theorem

Geometrisation of Chaplygin’s reducing multiplier theorem

A V Bolsinov, A V Borisov, I S Mamaev Dept. of Mathematical Sciences, Loughborough University, Loughborough,
LE11 3TU, UK
Udmurt State University, Izhevsk, Russia A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Moscow, Russia
Abstract

We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie-Poisson -bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.

ams:
37J60, 37J35, 70E18, 53D17

, ,

Introduction

In [21] S. A. Chaplygin found a special class of systems with two degrees of freedom which can be reduced to a Lagrangian and thus Hamiltonian form by a suitable change of time , where  is a reducing multiplier depending on the coordinates. As an illustration, he considered the problem of motion of the so-called Chaplygin sleigh, which can be integrated by the Hamilton – Jacobi method using the reducing multiplier method proposed by himself. Afterwards it was shown that a number of systems in nonholonomic mechanics can also be represented in the form of Chaplygin systems or generalised Chaplygin systems [5], and thereby are conformally Hamiltonian [8, 5, 18, 2, 13]. Thus, the reducing multiplier method is one of the most effective methods for explicit Hamiltonisation of dynamical systems.

From today’s perspective, the reducing multiplier theory is a method for finding one of the most important tensor invariants [10] of a dynamical system — the Poisson structure [5]. At the same time, the application of this method requires rewriting the equations of motion in local coordinates, which usually involves extremely cumbersome calculations. In this paper we develop the Chaplygin method for one class of systems frequently discussed in nonholonomic mechanics, which allows to achieve their Hamiltonisation in a much simpler way. We shall not dwell here on the derivation of equations of motion for nonholonomic mechanics. A fairly detailed treatment of this can be found in [4].

1 Generalised Chaplygin systems

We recall that according to [5], a generalised Chaplygin system is a mechanical system with two degrees of freedom whose equations of motion can be written as

 ddt(∂L∂˙q1)−∂L∂q1=˙q2S,ddt(∂L∂˙q2)−∂L∂q2=−˙q1S,S=a1(q)˙q1+a2(q)˙q2+b(q), (1)

where is a function of generalised coordinates and velocities , which we may call the Lagrangian of the system. It is straightforward to verify that this system admits an energy integral of standard form

 E=∑i∂L∂˙qi˙qi−L. (2)

Remark. A usual Chaplygin system can be obtained by a special choice of the function (a fortiori ) [21]. A somewhat different generalisation of the Chaplygin systems is proposed in [7, 9].

If there is an invariant measure with density depending only on the coordinates, the system can be represented in conformally Hamiltonian form [5] (for this was shown by S. A. Chaplygin [21]). To show this, we use the Legendre transform for the initial system (1):

 Pi=∂L∂˙qi,H=∑iPi˙qi−L∣∣˙qi→Pi.

Then the equations of motion (1) can be recast as

 ˙qi=∂H∂Pi,˙P1=−∂H∂q1+∂H∂P2S,˙P2=−∂H∂q2−∂H∂P1S,S=a1(q)˙q1+a2(q)˙q2+b(q)=A1(q)P1+A2(q)P2+B(q). (3)

Here coincides with the energy integral (2) expressed in terms of the new variables.

Now assume that the system admits an invariant measure with density depending only on the coordinates:

 μ=N(q)dP1dP2dq1dq2. (4)

In this case the Liouville equation for reduces to

 ˙q1(1N∂N∂q1−A2(q))+˙q2(1N∂N∂q2+A1(q))=0,

and since depends only on the coordinates, each of the brackets must vanish separately:

 1N∂N∂q1−A2(q)=0,1N∂N∂q2+A1(q)=0. (5)

Let us now make the change of variables

 Pi=piN(q),i=1,2.

Denote the Hamiltonian in the new variables as . Then the following relations hold for the derivatives

 ∂H∂Pi=N∂¯¯¯¯¯H∂pi,∂H∂qi=∂¯¯¯¯¯H∂qi+1N∂N∂qi(∂¯¯¯¯¯H∂p1p1+∂¯¯¯¯¯H∂p2p2).

Substituting them into (3) and using (5), we obtain

 ˙qi=N(q)∂¯¯¯¯¯H∂pi,˙p1=N(q)(−∂¯¯¯¯¯H∂q1+N(q)B(q)∂¯¯¯¯¯H∂p2),˙p2=N(q)(−∂¯¯¯¯¯H∂q2−N(q)B(q)∂¯¯¯¯¯H∂p1).

Thus, the following result holds.

Theorem 1

If the system (3) admits an invariant measure of the form (4), it can be represented in conformally Hamiltonian form

 ˙qi=N(q){qi,¯¯¯¯¯H},˙pi=N(q){pi,¯¯¯¯¯H},i=1,2,

where the Poisson brackets are given by

 {qi,pj}=δij,{qi,qj}=0,{p1,p2}=N(q)B(q).

Proof. The proof is a straightforward verification of the Jacobi identity.

2 The Chaplygin system on T∗S2

We now consider a system which is described by means of two three-dimensional vectors and and whose equations of motion are

 ˙M=(M−Sγ)×∂H∂M+γ×∂H∂γ,˙γ=γ×∂H∂M, (6)

where the “Hamiltonian”  is an arbitrary function (quadratic and non-degenerate in ) and is a function linear in :

 S=(K(γ),M)=K1(γ)M1+K2(γ)M2+K3(γ)M3.

It can be proved by a straightforward verification that the system (6) always admits three integrals of motion:

 F1=γ2,F2=(M,γ),F3=H(M,γ).

Without loss of generality we can set , so that equations (5) govern the dynamical system on the family of four-dimensional manifolds

 M4c={M,γ∣γ2=1,(M,γ)=c},

each of which is diffeomorphic to .

If the entire set of variables is denoted as , then equations (6) can be represented in the skew-symmetric form

 ˙x=P0∂H∂x, P0=(MΓΓ0)−S(x)(ccΓ000),

Here the first term is a standard Poisson structure corresponding to the Lie algebra . Moreover, additionally satisfies the equations

 P0∂F1∂x=0,P0∂F2∂x=0.

As above, assume that (6) admits an invariant measure with density depending only on :

 μ=ρ(γ)dMdγ. (7)

In this case the Liouville equation for the vector field defined by the system (6) can be represented as

 divρV=(∂H∂M,ργ×K−γ×∂ρ∂γ)=0.

Hence, owing to non-degeneracy of the Hamiltonian in , we obtain the vector equation

 (1ρ∂ρ∂γ−K)×γ=0. (8)

Using this relation, we can prove by direct computation

Proposition 1

If satisfies equation (8), then the tensor satisfies the Jacobi identity and therefore is a Poisson structure on .

Thus, we finally obtain

Theorem 2

If the system (6) admits an invariant measure (7) with density depending only on , it can be represented in the conformally Hamiltonian form

 ˙x=ρ(γ)P(x)∂H∂x,

where  is a Poisson structure of rank 4 with the Casimir functions

 F1=γ2,F2=(M,γ).

Equation (8) can be solved for the vector as follows:

 K=ρf(γ)γ+1ρ∂ρ∂γ,

where  is an arbitrary function. Thus, we have naturally obtained a special class of Poisson structures on the space , which can be written as

 P=1ρ(MΓΓ0)−(1ρ2∂ρ∂γ+f(γ)γ,M)(Γ000). (9)

Remark. If we add a term of the form

 Φ(γ)(Γ000)

to the bracket (9), where  is an arbitrary function, then the Jacobi identity will still hold. We use such a modification of (9) below, see (20).

We give two examples.

The problem of the Chaplygin ball on a plane [22] describing the rolling of a balanced dynamically asymmetric ball without slipping on a horizontal plane.

In appropriate variables the equations of motion can be represented in the form (6), see [14, 3, 4] with

 H1=12((AM,M)+(AM,γ)2D−1−(γ,Aγ))+U1(γ),S=(AM,γ)D−1−(Aγ,γ), (10)

where ,  is a constant diagonal matrix. The ball’s angular momentum relative to the point of contact is expressed in terms of the physical variable , angular velocity, by the formula

 M=A−1ω−D(ω,γ)γ,ω=A(M+Sγ), (11)

and the integral can take arbitrary values.

The density of the invariant measure (7) of the system and the function for the bracket (9) has the form

 ρ=1√D−1−(γ,Aγ),f(γ)=0.

The Veselova system [15, 16, 19] governing the dynamics of a body with a fixed point subject to the nonholonomic constraint , where  is the angular velocity of the ball and  is a unit vector fixed in space.

In the body-fixed frame, the equations of motion can be represented in the form (6), see [6] with

 H2=12⎛⎝(M,ˆAM)−((ˆA−E)M,γ)2(ˆAγ,γ)⎞⎠+U2(γ),S=−((ˆA−E)M,γ)(ˆAγ,γ), (12)

where  is the constant matrix inverse to the tensor of inertia, and the angular momentum is expressed in terms of the angular velocity of the body as follows

 M=ˆA−1ω+((ˆA−1−E)ω,γ)γ,ω=ˆA(M−Sγ), (13)

where the area integral coincides with the constraint equation:

 (M,γ)=(ω,γ)=b.

The density of the invariant measure (7) and the function coincide in this case:

 ρ(γ)=f(γ)=1√(γ,ˆAγ).

A Lagrangian representation for after a change of time was obtained in [8], the corresponding conformally Hamiltonian representation in [6], and another conformally Hamiltonian representation was found in [5].

If the potential for these systems is not zero, then, as a rule, the corresponding equations of motion turn out to be nonintegrable so that this Hamiltonisation method is essentially different from that used in [1], where the existence of a complete set of first integrals was required.

We also note that if one makes a change of the parameters and the potential in the Chaplygin ball problem:

 A=D−1(E−ˆA),U1(γ)=D−1U2(γ),

then we find that the Hamiltonian (10) becomes

 H1=D−12(M,M)−D−1H2, (14)

and the Poisson structure of the Chaplygin ball is transformed into a Poisson structure of the Veselova system. Consequently, these two systems are defined on the same Poisson manifold [20], and their Hamiltonians are related by (14). If , , then the function is an integral for the both systems, which implies that their trajectories turn out to be rectilinear windings (transverse to each other) on the same invariant tori [19].

3 Reduction to the e(3)-bracket

Introducing new notation , we can rewrite the Poisson structure (9) in a shorter form that is more convenient for further analysis

 P=g(MΓΓ0)+(∂g∂γ−f⋅γ,M)(Γ000). (15)

Let us examine the family of such Poisson structures in more detail. First of all, we see that this family is parametrised by two arbitrary functions and and we will denote the corresponding Poisson structures by . Notice that all possess the same Casimir functions and .

For simplicity we confine our attention to the physical case , that is, we restrict all the objects to the five-dimensional (Poisson) manifold .

One of our goals is to find out to what canonical form these Poisson structures can be reduced. First of all, we note that the symplectic leaves of are all diffeomorphic to the cotangent bundle to the sphere . From the explicit form (15) of the Poisson structure it may be inferred that the symplectic structure on each leaf will be the sum of the canonical form and a magnetic term, that is, a closed 2-form on the sphere. By the Moser theorem [12], such forms are parametrised up to a symplectomorphism by one single number, namely . Thus, for each Poisson structure we have a one-parameter family of symplectic leaves whose type is also defined by exactly one parameter. This observation leads us to the conjecture that by “redistributing”, if necessary, the symplectic leaves and then by applying a certain symplectomorphism to each single symplectic leaf, we can transform any Poisson structure to any other .

Remark. On the zero level , the Poisson structure (15) is reduced to the canonical -bracket by a very simple transformation [5]:

 (M,γ)↦(g−1(γ)M,γ).

Thus, for the Chaplygin ball we have:

 (M,γ)↦((D−1−(γ,Aγ))−1/2M,γ),

and for the Veselova system:

We start by describing a class of natural transformations which preserve the form of , but change the parameters and . Consider the transformations of the form

 (M,γ)↦(˜M,γ),˜M=A(γ)M, (16)

where is a linear operator in whose components depend on .

Proposition 2

For each point , consider the orthogonal decomposition , where is the projection of onto the vector , and is the projection of onto the plane perpendicular to , i.e., the tangent plane . Let

 ˜M=α(γ)M′+cM′′+M′′×h(γ),

where is a constant, is an arbitrary scalar function, and is an arbitrary vector function of . Then the transformation (16) sends to a Poisson structure of the same kind with parameters

 ˜g=αg,˜f=α2cf+(αc−1)(˜g−(γ,∂˜g∂γ))+1c(γ,˜g∂α∂γ+˜g2curl(h˜g)). (17)

The proof of Proposition 2 is a straightforward verification and we confine ourselves to commenting on the geometric meaning of the transformation used in this proposition. Consider the orthonormal basis related to the vector in the space . Namely, and are two orthonormal vectors lying in a tangent plane to the unit sphere at point , and is the normal vector to this sphere at the same point, i.e., . In this basis the matrix of has the form

 A=⎛⎜⎝α0a0αb00c⎞⎟⎠

where , and depend on , and is constant.

This is exactly the general form of the transformation which satisfies our requirements. Indeed, the Casimir function should be mapped to itself with possible multiplication by some constant . Therefore, the plane defined by the equation is sent to itself, and in the orthogonal direction the transformation is a dilatation with ratio independent of . These conditions completely define the last row of the matrix .

Furthermore, the relations can be formally rewritten in vector form as . Since their form must remain the same, we obtain the condition

 g(A(γ)M)×γ=˜gM×γ.

This means that on the tangent plane the operator must act as multiplication by some number (depending on ). There are no restrictions on the elements and , they are given by the vector function  (this function itself has 3 components, but only two of them are significant, since nothing is changed by adding to any vector proportional to ).

Notice that the set of transformations described in Proposition 2 forms a group (which is, of course, infinite-dimensional, since its parameters contain arbitrary functions and ). It is easily verified that performing successively two transformations with parameters and is equivalent to the transformation with parameters . The above-mentioned rule specifies a group binary operation, which simply copies the matrix multiplication:

 (α2h20c2)(α1h10c1)=(α1α2h1α2+h2c10c1c2)

This group acts in a natural way on the family of Poisson structures or, which is the same, on the space of parameters . The above relations (17) can be understood as explicit formulae for this action. If the action is formally denoted by , then, as is easily verified by successively performing two transformations, it satisfies the standard action rule. Namely, if

 (˜g,˜f)=Ψ(α1,c1,h1)(g,f)and(˜˜g,˜˜f)=Ψ(α2,c2,h2)(˜g,˜f),

then

 (˜˜g,˜˜f)=Ψ(α1α2,c1c2,h1α2+h2c1)(g,f).

For an explicit verification of this fact it is convenient to rewrite (17) as

 ˜g=αg,˜f=α2c(f+g−(γ,∂g∂γ))−(˜g−(γ,∂˜g∂γ))+˜g2c(γ,curl(h˜g)).

Now the verification presents no difficulty.

From the viewpoint of group theory it would now be natural to ask the question: what are the orbits of this action? In other words, we want to understand which Poisson structures may be transferred to each other by the above-mentioned transformations. The answer turns out to be very simple: the action described above has one single orbit, i.e., all Poisson structures in this family are equivalent to each other. In particular, the following theorem holds:

Theorem 3

Every Poisson structure of the form (15) on the level is isomorphic to the standard Lie-Poisson structure related to the Lie algebra .

Proof. It is sufficient to choose parameters in (17) in such a way that and . The first condition immediately defines the function , namely, . After that the second condition reduces to

 α2cf+(αc−1)+1c(γ,∂α∂γ)+1c(γ,curlh)=0

or, equivalently,

 α2f+α+(γ,∂α∂γ)−c+(γ,curlh)=0,

where the constant and the vector function are the unknowns. This equation can now be rewritten as

 (γ,curlh)=F(γ)+c, (18)

where is a given function. Notice that (18) has to be fulfilled only on the unit sphere . The conditions for solving the equations of this form are well known. In the differential-geometric sense this equation simply means that we are looking for an antiderivative of the -form on the unit sphere, where  is the standard area form. Such a 1-form can be found if and only if . This condition can always be achieved by choosing a constant .

Remark. In a similar manner, the bracket can be reduced to the standard form on the whole space , i.e., without the additional restriction . To that end, we have to extend the class of transformations by assuming that depends on . Since is a Casimir function, may be treated, as before, as a constant and hence the formulae do not essentially change. The conditions for solvability of the equation remain the same, but now they have to be verified on the spheres of all radii. As before, we are able to ensure that they are satisfied, since the necessary constants can now be chosen depending on the square of the radius .

For the Chaplygin ball problem, the function in Eq. (18) has the form

 F(γ)=−D−1(D−1−(γ,Aγ))3/2.

The solutions of Eq. (18) for the unknowns and can be expressed in this case in terms of complete and incomplete elliptic integrals. Thus, although theoretically it is not difficult to prove reducibility of to the -bracket, in practice the resulting transformation can turn out to be extremely unwieldy and non-algebraic.

4 Generalisation to the case of a gyrostat

In this section we consider the dynamical systems obtained by adding a rotor with constant gyroscopic momentum to the Chaplygin ball and a rigid body in the Veselova problem. A detailed derivation of the equations of motion for compound bodies can be found in books [23, 24, 25].

The new equations with gyrostatic terms take the following form

 ˙M=(M+k−Sγ)×∂H∂M+γ×∂H∂γ,˙γ=γ×∂H∂M, (19)

where the new “Hamiltonian” and function may now depend on the gyrostatic momentum as a parameter, but preserve their original structure as in Section 2. In particular,

 S=1g(−∂g∂γ+f(γ)γ,M)+1gΦ(γ) (20)

for some smooth functions and .

A direct calculation shows that this system remains conformally Hamiltonian. Namely, (19) can be rewritten as

 ˙x=g−1Pk(x)∂H∂x,

with the Poisson structure of a more general form

 Pk(x)=g(MkΓΓ0)−gS(Γ000),Mk=⎛⎜⎝0−M3−k3M2+k2M3+k30−M1−k1−M2−k2M1+k10⎞⎟⎠, (21)

where  is a complete set of variables.

The Jacobi identity for is fulfilled, and the Casimir functions are

 F1=γ2,F2=(M+k,γ).

The new expressions for and presented below can be obtained by using the methods developed in [23, 24, 25]. We omit this computation.

For the Chaplygin ball, the vector is still expressed in terms of the angular velocity by means of (11), and the Hamiltonian (10) also remains the same. For the bracket (21) we set

 g=√D−1−(γ,Aγ),f(γ)=0,Φ(γ)=0.

In other words, all the ingredients remain unchanged except for the additional terms involving in the bracket (21). Thus, to obtain the gyrostatic generalisation of the Chaplygin ball we simply need to replace by in (15).

For the Veselova system, when a gyrostat is added, the situation becomes less trivial and the relations (13) as well as and given by (12) need to be modified. As before, we shall assume that , where the coefficient can be found from the condition

 (M+k,γ)=(ω,γ).

We obtain

 M=ˆA−1ω−((ˆA−1−E)ω+k,γ)γ,ω=ˆA(M−Sγ),S=(ˆAM−M−k,γ)(ˆAγ,γ).

Here coincides with the corresponding function in the bracket (21) provided that is given as

 g=√(ˆAγ,γ).

In this case the Hamiltonian reads

 H=12((ˆAM,M)+(ˆAM−M−k,γ)2(ˆAγ,γ)).

It turns out that this modified Veselova system with gyrostatic terms still admits one additional integral of the form

 F3=(M+k,M+k).

Thus, this new system is conformally Hamiltonian and integrable. Its dynamics can be further analysed by the standard methods.

Conclusion and discussion

We have obtained an invariant (independent of the choice of local coordinates on ) conformally Hamiltonian representation of generalised Chaplygin systems on using a degenerate Poisson structure of rank 4 in the six-dimensional space and have shown that this structure is a deformation of the standard Lie – Poisson bracket in corresponding to the Lie algebra .

As applications, we have considered two nonholonomic systems: the Chaplygin ball and Veselova problem. In this approach (after a suitable change of parameters) they turn out to be integrable conformally Hamiltonian systems on the same Poisson manifold with the same set of first integrals. The above conformally Hamiltonian representation has been generalised to the case of adding a gyrostat (although in this case there is no analogy between these systems any more).

To the best of our knowledge, the conformally Hamiltonian description for the Veselova system with and the integrability of its gyrostatic generalisation were unknown before and are presented in this paper for the first time.

This paper poses a number of questions related primarily to nonholonomic systems.

1. Can the above approach be used to obtain a conformally Hamiltonian description for an integrable generalisation of the Chaplygin ball rolling on a spherical base (BMF-system) found in [2, 13, 20]?

2. Poisson brackets of a quite similar type are encountered in examples but with a Casimir function linear in different from  [1]. It would be interesting to find out whether such brackets can be reduced to the standard Poisson-Lie bracket on using the technique described above.

3. Since the Chaplygin ball problem without potential (i.e., ) is integrable on the whole space , Theorem 3 allows us to obtain a globally integrable Hamiltonian system on , i.e., for all values of the area constant . As is well known, this circumstance may be interpreted as integrability of a natural system with a magnetic field whose additional integral is quadratic in momenta. The issue of description of all such systems was actively discussed in the literature. It would be interesting to interpret the system thus obtained in the context of recent classification results by V. Marikhin and V. Sokolov [11, 17].

Acknowledgments

The work of Alexey V. Borisov was carried out within the framework of the state assignment to the Udmurt State University “Regular and Chaotic Dynamics”. The work of Ivan S. Mamaev was supported by the RFBR grants 13-01-12462-ofi m.

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