Integrable nonholonomic geodesic flows on compact Lie groups

arXiv:math-ph/0408037 v1 24 Aug 2004 Integrable nonholonomic geodesic flows on compact Lie groups ∗ Yuri N. Fedorov Department of Mathematics and Mechanics Moscow Lomonosov University, Moscow, 119 899, Russia e-mail: [email protected] and Departament de Matem`atica I, Universitat Politecnica de Catalunya, Barcelona, E-08028 Spain e-mail: Yuri.Fedoro[email protected] and Boˇzidar Jovanovi´c Mathematical Institute, SANU Kneza Mihaila 35, 11000, Belgrade, Serbia e-mail: [email protected] August 24, 2004 Abstract This paper is a review of recent results on integrable nonholonomic geodesic flows of left–invariant metrics and leftand right–invariant constraint distributions on compact Lie groups. Contents 1 Introduction 2 1.1 Nonholonomic Geodesic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Chaplygin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contents of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 LL Systems 7 2.1 Euler–Poincar´e–Suslov Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Some Integrable Cases of EPS Equations . . . . . . . . . . . . . . . . . . . . . 9 2.3 Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 LR Systems 14 3.1 LR Systems as Generalized Chaplygin Systems . . . . . . . . . . . . . . . . . 14 3.2 Veselova Problem, an Integrable Geodesic Flow on the Sphere and the Neumann Problem 16 3.3 Reconstructed Motion on D. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Veselova Problem with Integrable Potentials and the Maupertuis Principle . . 22 ∗AMS Subject Classification 37J60, 37J35, 70H45 1 4 L+R Systems 24 4.1 Definition and Invariant Measure of L+R Systems . . . . . . . . . . . . . . . 24 4.2 The spherical Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Limits of L+R Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 Introduction This paper is a review of recent results on integrable flows on compact Lie groups under nonholonomic constraints. We mostly follow papers [24, 30, 31, 26], trying to present their results within a unified framework. Furthermore, some new examples of integrable nonholonomic systems are given. 1.1 Nonholonomic Geodesic Flows We start with basic definitions and settings. Let (Q, ds2) be n–dimensional Riemannian manifold Qwith a nondegenerate matric ds2and a Levi–Civita connection ∇,Dbe a nonintegrable k–dimensional distribution on the tangent bundle T Q. A smooth path γ(t)∈ Q, t ∈∆ is called admissible (or allowed by constraints) if the velocity ˙γ(t) belongs to Dγ(t) for all t∈∆. There are two approaches to define geodesic lines among admissible paths: by induced connection as “straightest” lines and by the variation principle as “shortest” lines. We shall deal with the first approach which arises from mechanics. The admissible path γ(t) is called a nonholonomic geodesic if it satisfies the equations π(∇˙γ(t)˙γ(t)) = 0,(1.1) where π:TqQ→Dq,q∈Qis the orthogonal projection. Equivalently, we can introduce the Lagrangian function l=1 2(K˙q, ˙q), where Kis the metric on Qalso regarded as a mapping K:T Q →T∗Q. Let q= (q1,...,qn) be some local coordinates on Q. The trajectory of the system q(t) that satisfies the constraints is a solution to the Lagrange–d’Alambert equations ∂l ∂q −d dt ∂l ∂˙q, η=X i∂l ∂qi−d dt ∂l ∂˙qiηi= 0,for all η∈Dq.(1.2) One can also write the Lagrange-d’Alambert equations as a first-order system on the (n+k)-dimensional constraint submanifold M=K(D) of the cotangent bundle T∗Q. Let Dbe locally defined by ρ=n−kindependent 1-forms αi Dq={ξ∈TqQ, (αj q, ξ) = X i αj iξi= 0, j = 1,...,ρ}. Then Mis locally given by the equations (αi q, K−1 qp) = 0, i= 1, . . . , ρ. Let pi=∂l/ ˙qi, i= 1,...,n be momenta which together with qprovide canonical coordinates on T∗Q. Let h(q, p) = 1 2(p, K−1 qp) be the Hamiltonian function (the usual Legendre transformation of L). The equations (1.2) are equivalent to ˙pi=−∂h(q, p) ∂qi + ρ X i=1 λjαj(q)i,˙qi=∂h(q, p) ∂pi , i = 1,...,n, (1.3) where Lagrange multipliers are chosen such that the solutions (q(t), p(t)) belong to M. As for the Hamiltonian systems, the Hamiltonian function is always the first integral of the system. There is also a nonholonomic version of the Noether theorem (see [38, 2, 24, 6]). 2 The Noether theorem. Suppose that a Lie group Gacts on the configuration space Qand that the action is naturally extended to T Q and T∗Q. The momentum mappings Ψl:T Q →g∗and Ψ∗:T∗Q→g∗are defined by Ψl(q, ˙q|ξ) = ∂l ∂˙q, ξQ= (Kq˙q, ξQ),Ψ∗(q, p |ξ) = (p, ξQ),(1.4) where ξQis the vector field on Qassociated to the action of one-parameter subgroup exp(tξ), ξ∈g=TIdG. Theorem 1.1. Assume that ξQis a section of the distribution Dand the one-parameter subgroup exp(tξ)preserves l(or h). Then Ψl(ξ)is the first integral of the system (1.2), or equivalently, Ψ∗(ξ)is the first integral of (1.3). Invariant measure and integrability. The equations (1.3) are not Hamiltonian. This is why it is still not clear how to define the notion of complete integrability for nonholonomic systems (see [4]). However, in some cases they have an invariant measure, a rather strong property, which puts the system close to Hamiltonian systems. In particular, if apart from the Hamiltonian there exist dim M−3 additional independent integrals, then, by the Euler– Jacobi theorem, the solutions of (1.3) can be found by quadratures. The importance of an invariant measure for integrability of nonholonomic systems was indicated by Kozlov in [39], where various examples were discussed (see also [2]). Namely, consider a non-Hamiltonian system ˙x=f(x), x ∈Rm,(1.5) having an invariant measure µ(x)dx and m−2 first integrals F1(x), . . . , Fm(x). If the latter are independent on the invariant set Mc={x∈Rm, Fi(x) = ci, i = 1,...,m−2}, then Mcis a two-dimensional submanifold and the flow on Mchas also an invariant measure. Then, accoring to the Euler–Jacobi theorem, solutions of (1.5) lying on Mccan be found by quadratures. Moreover, if Lcis a compact connected component of Mcand f(x)6= 0 on Lc, then, if orientable, Lcis diffeomorphic to two-dimensional torus. According to Kolmogorov’s theorem on reduction of differential equations with a smooth invariant measure on a torus ([37]), one can find angular coordinates ϕ1, ϕ2on Lc, in which the reduction of equations (1.5) takes the form similar as in the Liouville theorem: ˙ϕ1=Ω1 Φ(ϕ1, ϕ2),˙ϕ2=Ω2 Φ(ϕ1, ϕ2), where Ω1,Ω2depend on the constants of motion c1,...,cm−2only and Φ is a smooth positive 2π–periodic function in ϕ1, ϕ2, the density of the induced invariant measure on Lc. Therefore, it is natural to call the system (1.5) completely integrable if it can be integrated by the Jacobi theorem; or, more generally (see [54, 55]), if the phase space is almost everywhere foliated by invariant tori Tk{ϕ1,...,ϕk}with the dynamics of the form ˙ϕ1=Ω1 Φ(ϕ1,...,ϕk), ..., ˙ϕk=Ωk Φ(ϕ1,...,ϕk).(1.6) The above definition of complete integrability is slightly different from the definition of complete integrability of non-Hamiltonian systems given in [8, 60]. Namely, here we have quasi-periodic motions after the time substitution dτ = Φ−1(ϕ)dt. The existence of an invariant measure for smooth dynamical systems and for a class of nonholonomic systems with symmetries is studied in [40] and [59], respectively. Various mechanical examples with an invariant measure can be found in [12]. The authors 3 of the paper [54, 55] constructed nonholonomic systems on unimodular Lie groups with right-invariant nonintegrable constraints and a left-invariant metric (so called LR systems), and showed that they always possess an invariant measure, whose density can be effectively calculated. In particular, the motion of a rigid body around a fixed point under a nonholonomic constraint (projection of the angular velocity to the fixed vector in space is constant) is described by an integrable LR system ([54]). Similar integrable problems on Lie groups with left–invariant constraints are studied in [26, 30, 31]. Also, an important example of an integrable nonholonomic mechanical system, the problem of rolling of a homogeneous ball on a surface of revolution (the Routh problem), was treated in detail in [27, 57]. 1.2 Chaplygin Systems Another approach to the integrability of nonholonomic systems is based on their reduction to a Hamiltonian form after an appropriate time rescaling. First, following [36] and [6], let us recall some basic facts about the Chaplygin systems. Let (Q, l, D) be a nonholonomic system with a Lagrangian lof the natural mechanical type, with kinetic energy that correspods to the metric ds2and the potential function v. Assume that there is a bundle structure π:Q→Nwith the base manifold Nand let the map πbe a submersion, such that TqQ=Dq⊕Vqfor all q. Here Vqis the kernel of Tqπcalled the vertical space at q. Then the distribution Dcan be seen as a collection of horizontal spaces of the Ehresmann connection associated to π:Q→N. Given a vector Xq∈TqQ, there is a decomposition Xq=Xh q+Xv q, where Xh q∈Dq,Xv q∈Vq. The curvature of the connection is the vertical valued 2-form Bon Qdefined by B(Xq, Yq) = −[¯ Xh q,¯ Yh q]v q, where ¯ Xand ¯ Yare smooth vector fields on Qobtained by extending of Xqand Yq. By applying the Ehresmann connection the Lagrange–d’Alambert equations (1.2) can be represented in the form (see [6]) ∂lc ∂q −d dt ∂lc ∂˙q, η=∂l ∂˙q, B( ˙q, η),for all η∈Dq,(1.7) where lc(q, ˙q) = l(q, ˙qh) is the constrained Lagrangian. Now, suppose that π:Q→N=Q/Gis a principal bundle with respect to the left action of a Lie group G, and Dis a principal connection, i.e., Dis a G-invariant distribution. Let the Lagrangian lbe also G-invariant, i.e., Gacts by isometries on Riemannian manifold (Q, ds2) and vis a G–invariant function. Then the constrained Lagrangian lcinduces a well defined reduced Lagrangian L:T Q →Rvia identification T N ≈D/G. The reduced Lagrangian Lis of the natural mechanical type as well. Its kinetic energy is given by metric ds2 Dand its potential energy will be denoted by V. Under the above assumtions, equations (1.7) are G-invariant and induce reduced Lagrange– d’Alambert equations on the tangent bundle T N, ∂L ∂q −d dt ∂L ∂˙q, η= Σ( ˙q, η),for all η∈TqN. (1.8) Here Σ is semi-basic two-form given by the right hand side of (1.7) and q= (q1,...,qk) are some local coordinates on the base space N. From (1.4) we see that Σ depends on the curvature of the connection Dand on the momentum mapping Φl. The system (Q, l, D, G) is referred to as a (generalized) Chaplygin system (see [36, 6]), as a generalization of classical Chaplygin systems with Abelian symmetries [17]. 4 Remark 1.1. Note that horizontal and vertical spaces do not need to be orthogonal with respect to the metric ds2. In fact, if Dis ds2-orthogonal to the leaf of G-action, then D will be an invariant submanifold of the nonconstrained geodesic flow of the metric ds2, and the right hand sides of equations (1.7) will be zero. In this case, ds2 Dcoincides with the submersion metric induced from ds2. Chaplygin’s reducing multiplier. Let pi=∂L/∂ ˙qi,i= 1, . . . , k be momenta, gij the metric tensor of ds2 Dand gij the dual metric on T∗N. Then the reduced Lagrangian has the form L(q, ˙q) = 1 2Pgij ˙qi˙qj−V(q). We also introduce the Hamiltonian function H(q, p) = 1 2Pgijpipj+V(q). The reduced system (1.8) can be rewritten as a first-order dynamical system on T∗N: ˙qi=∂H ∂pi ,˙pi=−∂H ∂qi + Πi(q, p), i = 1,...,k. (1.9) The functions Πiare quadratic in momenta and can be regarded as non-Hamiltonian perturbations of the equations of motion of a particle on N. Let Ω = Pdpi∧dqibe the standard symplectic form on T∗N. The equations (1.9) have an invariant measure fΩkif Pi∂(f˙qi) ∂qi+∂(f˙pi+fΠi) ∂pi= 0. Since the finction fdepends only on the coordinates q, this is equivalent to condition d(ln f) + α= 0,(1.10) where the one-form αis given by Pi ∂Πi ∂pi|˙q=gp = (α, ˙q). Remark 1.2. The paper [50] (see also [13]) contains a nontrivial observation about the density of the invariant measure, which in our terms reads as follows. Suppose that system (1.9) has an invariant measure with density f(q, p) in the case of absence of potential (V(q) = 0). Then one can check that the function f0(q) = f(q, 0) is also a solution of (1.10). In other words, if the reduced system (1.9) has an invariant measure for V= 0, one can take this measure to be of the form f(q)Ωk. Then, since (1.10) does not depend on the potential, the reduced system (1.9) has the same invariant measure in the presence of a potential field V(q) as well. Now consider time substitution dτ =N(q)dt, where N(q) is a differentiable nonvanishing function on Q, and denote q′=dq/dτ. Then we have the following commutative diagram T N{q, ˙q}q′= ˙q/N(q) −−−−−−−→ T N{q, q′} p=g˙q  y  y˜p=N2gq′ T∗N{q, p}˜p=Np −−−−→ T∗N{q, ˜p}. The Lagrangian and Hamiltonian functions in the coordinates {q, q′}and {q, ˜p}take the form L∗(q, q′) = 1 2XN2gijq′ iq′ j−V(q), H∗(q, ˜p) = 1 2X1 N2gij ˜pi˜pj+V(q). There is a remarkable relation between the existence of an invariant measure of the reduced system (1.9) and its reducibility to a Hamiltonian form (see [26]). Theorem 1.2. 1). Suppose that after the time substitution dτ =N(q)dt the equations (1.9) become Hamiltonian, q′ i=∂H∗ ∂˜pi ,˜p′ i=−∂H∗ ∂qi .(1.11) 5 Then the function f(q) = N(q)k−1satisfies the equation (1.10), i.e., the original system (1.9) has the invariant measure with density f(q). 2). For k= 2, the above statement can also be inverted: the existence of the invariant measure with the density N(q)implies that in the new time dτ =N(q)dt, the system (1.9) gets the Hamiltonian form (1.11). In nonholonomic mechanics the factor Nis known as the reducing multiplier, item 2) of this theorem is referred to as Chaplygin’s reducibility theorem (see [16, 17] or section III.12 in [45]). Notice that for k > 2, the multiplier N(q) and the density of the invariant measure of system (1.9) do not coincide. Also, the existence of the multiplier do not depends on the potential V. There are many examples of the Chaplygin reducing multiplier for k= 2. Since many conditions on the metric and constraints are imposed, until recently there were no nontrivial examples of multidimensional systems, appart of several examples for k= 3,4 with the property that factor N(q) depends only on one coordinate, that are reducible to a Hamiltonian form by the Chaplygin procedure ([45, 20, 28, 44]). As an alternative, in the reduction of Chaplygin systems one can use the symplectic (or Poisson) framework (see [50, 3, 13, 14]). Such systems can be represented in a Hamilton-like form with respect to an nondegenerate (almost-symplectic) 2-form, which however may be not closed Namely, let Ξ be the Legandre transformation of the semi-basic form Σ. Then one can write (1.9) as Ωnh(XH,·) = dH(·),where Ωnh = Ω + Ξ. In this framework, the Chaplygin multiplier is a function Nsuch that the form ˜ Ω = NΩnh is closed. Then, after rescaling Y=X/N, we obtain the Hamiltonian system ˜ Ω(Y, ·) = dH(·) (see [50, 27, 13, 21]). Contrary to the procedure described in Theorem 1.2, here the vector field Yhas no direct mechanical description. Recently, necessary and sufficient conditions for the existence of an invariant measure of the reduced system in case when the Lagrangian of the system is of a pure kinetic energy type are given in [13, 14]. 1.3 Contents of the Paper In section 2 we consider the systems with left–invariant metrics and left–invariant constraint distributions, so called LL systems. The equations of the motion reduce to the Euler–Poincar´e–Suslov equations on the corresponding Lie algebra. Although such equations generally are not Hamiltonian, their nice algebraic structure allows us to construct various integrable examples with an invariant measure. In section 3 we consider a class of LR systems (left–invariant metrics and right–invariant constraint distributions), which can be regarded as Chaplygin systems on the principle bundle G→Q=G/H,Hbeing a Lie subgroup. We show that, in contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Qalways possess an invariant measure. Then we study the case G=SO(n), when LR systems are multidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion, which admit a reduction to the system with an invariant measure on the (co)tangent bundle on the unit sphere Sn−1. For a special choice of the left-invariant metric on SO(n), we prove that under a time reparameterization, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere Sn−1. This provides a first multidimensional example of a nonholonomic system for which the celebrated Chaplygin reducibility theorem is applicable. Lastly, we present an explicit reconstruction of the motion on the group SO(n). 6 Finally, in section 4 we present another class of systems on an unimodular Lie group G, which always possess a non-trivial invariant measure and which are obtained as modifications of a geodesic flow on Gwith respect to a sum of a leftand a right-invariant metrics, so called L+R systems. It appears that a nonholonomic LR system on a group Gcan be obtained as a limit case of an appropriate L+R system on this group. As an example, we consider a nonholonomic mechanical system called the spherical support. 2 LL Systems 2.1 Euler–Poincar´e–Suslov Equations In this section we consider nonholonomic systems (G, l, D) with a left-invariant distributions Dand a left-invariant Lagrangians lthat describes left-invariant metrics on a compact connected Lie group G. Let g=TIdGbe the Lie algebra of G. In what follows we shall identify gand g∗by AdGinvariant scalar product h·,·i, and T G and T∗Gby bi-invariant metric on G. For clearness, we shall use the symbol ωfor the elements in gand the symbol xfor the elements in g∗∼ =g. Let d={ω∈G, hω, aii= 0, i = 1,...,ρ} ⊂ g be the restriction of the left-invariant distribution Dto the algebra, for some constant and linearly independent vectors aiin g. From the left invariance condition we have Dg=g·d. The distribution is nonintegrable if and only if dis not a subalgebra. Also, it is sufficient to give a Lagrangian at one point of the group, for instance the identity l(g, ˙g) = 1 2hIω, ωi, ω=g−1·˙g. Here I:g→gis a symmetric positive definite (with respect to h·,·i) operator. The Hamiltonian in the left-trivialization is given by H(x) = 1 2hA(x), xi,A=I−1. The corresponding left-invariant metric will be denoted by ds2 I. Let mbe the restriction of the constraint submanifold Mto g, that is m=I(d). Equations (1.3) are G–invariant and reduce to m, ˙x= [x, ∇H(x)] + ρ X i=1 λiai= [x, A(x)] + ρ X i=1 λiai,(2.1) where λiare Lagrange multipliers chosen such that xbelongs to m=I(d), i.e., such that ω= A(x) belongs to d:hA(x), aii= 0, i= 1,...,ρ. In other words, the following commutative diagram holds MPt −−−−→ M Λ  y  yΛ mPt −−−−→ m, (2.2) where Ptand Ptare phase flows of the nonholonomic geodesic flow and the system (2.1) respectively, and Λ maps g·x∈TgGto x∈g. Following [24], we shall call (2.1) the Euler–Poincar´e–Suslov (EPS) equations, as a generalization of the Suslov problem of the nonholonomic rigid body motion (see the example below). These equations have a quite different nature in comparison with the Euler–Poincar´e equations ˙x= [x, A(x)]. In particular, as indicated in [41], in the case of only one constraint ha, A(x)i= 0, they have a smooth invariant measure if and only if [a, A(a)] = µa. 7 Reconstruction of the motion on the group. In the Hamiltonian case, the integrability of the reduced system implies generally a non-commutative integrability of the original system, namely the phase space is foliated by invariant isotropic tori with quasi-periodic dynamic (see [60]). However there is no such analog in the nonholonomic setting. To reconstruct the motion (g(t),˙g(t)) on the whole phase space, we have to solve the kinematic equation g−1(t)·˙g(t) = ω(t) = A(x(t)), where x(t) are solutions of (2.1), i.e., to find all trajectories in Mthat projects to the given trajectory x(t) in m. In particular, if x(t) is a relative equilibrium (x(t) = x(t0) for all t) or if x(t) is a relative periodic orbit (x(t+T) = x(t) for all t), then the invariant set Λ−1({x(t), t ∈R})⊂ M is foliated by invariant tori of maximal dimension rank Gor rank G+ 1, respectively (e.g., see [27]). Multidimensional Suslov problem. The most natural example of LL systems is the nonholonomic Suslov problem, which describes the motion of an n-dimensional rigid body with a fixed point, that is, the motion on the Lie group SO(n), with certain left-invariant nonholonomic constraints. For a path g(t)∈SO(n), the angular velocity of the body is defined as the lefttrivialization ω(t) = g−1·g(t)∈so(n). The matrix g∈SO(n) maps a coordinate system fixed in the body to a coordinate system fixed in the space. Therefore, if e1= (e11,...,e1n)T...,en= (en1,...,enn)Tis the orthogonal frame of unit vectors fixed in the space and regarded in the moving frame, we have E1=g·e1, . . . , En=g·en, where E1= (1,0,...,0)T, . . . , En= (0,...,0,1)T. From the conditions 0 = ˙ Ei= ˙g·ei+ g·˙ei, we find that the vectors e1,...,ensatisfy the Poisson equations ˙ei=−ωei, i = 1,...,n. (2.3) The left-invariant metric on SO(n) is given by non-degenerate inertia operator I: so(n)→so(n). Then the Lagrangian of the free motion of the body reads l=1 2hIω, ωi, where now h·,·i denotes the Killing metric on so(n), hX, Y i=−1 2tr (XY ), X, Y ∈so(n). For a “physical” rigid body, Iωhas the form Iω +ωI, where Iis a symmetric n×nmatrix called mass tensor (see [24]). However, since we are interested mainly in nonholonomic geodesic flows, we shall consider other inertia operators as well. Recall that in the three-dimensional case, the Suslov problem describes the motion of a rigid body with the constraint: the projection of the angular velocity to a vector fixed in the moving frame (for example E3) is equal to zero [52, 2]. In other words, only infinitesimal rotations in the planes span(E1, E2) and span(E1, E3) are allowed. Hence, it is natural to define its n-dimensional analog as follows: only infinitesimal rotations in the fixed 2-planes spanned by (E1, E2), . . . , (E1, En) (i.e., in the planes containing the vector E1) are allowed. Following [24], one can relax these constraints by assuming that the angular velocity matrix has the following structure ω=         0··· ω1r··· ω1n . . ..... . .. . . −ω1r··· 0··· ωrn . . .. . .O −ω1n··· −ωrn         , 8 where Ois zero (n−r)×(n−r) matrix. This implies the left–invariant constraints hω, Ei∧Eji= 0, r + 1 ≤i < j ≤n. (2.4) As a result, the Suslov problem is described by the EPS equations d dt (Iω) = [Iω, ω] + X r<p<q≤n λpq Ep∧Eq,(2.5) together with Poisson equations (2.3). Here the components of the vectors e1,...,enplay the role of redundant coordinates on SO(n). Various integrable cases of the Suslov problem with additional potential fields and their multidimensional generalization are given in [34, 39, 2, 47] and [32], respectively. 2.2 Some Integrable Cases of EPS Equations EPS equations on symmetric pairs. Let hbe the subspace of the algebra gspanned by ai, i = 1,...,ρ. Consider the case when the tensor Apreserves the orthogonal decomposition g=h+d, i.e., A=Ah+Ad, where Ah:h→h,Ad:d→dare positive definite operators. Then m=I(d) = d, and we can write (2.1) in the following way ˙x= [x, Ad(x)]d, x ∈d,(2.6) where ξddenotes the orthogonal projection of ξ∈gto the subspace d(with respect to h·,·i). Equation (2.6) preserve the standard measure on d. Also the Hamiltonian function H(x) = 1 2hx, AD(x)iand the invariant F(x) = hx, xiare always first integrals of the system. Therefore, by the Jacobi theorem the equation (2.6) is always integrable for dim d≤4. Remark 2.1. Note that, in general, the invariant F(x) = hx, xiis not the integral of (2.1), although it is always an integral of non-constrained system. Namely, a first integral f(x) of the Euler–Poincar´e equations ˙x= [x, A(x)] is the integral of (2.1) if and only if the following condition holds X i λih∇f(x), aii|x∈m= 0.(2.7) In our case ∇F(x) = 2x,x∈m=dis orthogonal to hand therefore the invariant F(x) remains to be an integral. Example 2.1. Let kbe a subalgebra of gand wthe orthogonal complement of k. Suppose that (g,k) is a symmetric pair, i.e., the following conditions are satisfied: [w,w]⊂k,[k,k]⊂k,[k,w]⊂w. Then, in the special case d=w, we have [d,d]d= 0. Therefore all the solutions of equations (2.6) are constants. As a result, the solution of the original system on G(nonholonomic geodesic lines of the metric ds2 I) is given by the motion along one-parameter subgroups, g(t) = g0exp(tξ), ξ ∈d. This simplest situation occurs for the multidimensional Suslov equations (2.5) with r=n−1 and Iω=Iω +ωI, where I= diag (I1,...,In). Then h=so(n−1), (so(n), so(n−1)) is a symmetric pair, and Ipreserves the decomposition so(n) = d+so(n−1). Hence the solutions ω(t) are just constants. 9 Proposition 3.1. The reduced Lagrange–d’Alambert equation describing the motion of the LR system (G, l, D)has the form ∂L ∂q −d dt ∂L ∂˙q, ξ=hIΦ(q, ˙q),prg−1hg[Φ(q, ˙q),Φ(q, ξ)]i,(3.8) for all virtual displacements ξ∈TqQ, where prg−1hg:g→g−1hgis the orthogonal projection, and q=π(g). In addition, it appears that the reduced LR system (3.8) also possesses an invariant measure (note that a generic Chaplygin system does not have this property, see [13]). Namely, the following general statement holds (e.g., see [26]). Lemma 3.2. Suppose there is a compact group Gacting freely on a manifold Nwith local coordinates z and there is a G–invariant dynamical system ˙z=Z(z)on N. If this system has an invariant measure (which is not necessary G-invariant), then the reduced system on the quotient manifold N/Galso has an invariant measure. 3.2 Veselova Problem, an Integrable Geodesic Flow on the Sphere and the Neumann Problem Veselova problem. The most descriptive illustration of an LR system is the Veselova problem on the motion of a rigid body about a fixed point under the action of nonholonomic constraint (Ω, γ) = 0,(3.9) where Ω ∈R3is the angular velocity vector, γ∈R3is a unit vector, which is fixed in a space frame, and ( ,) denotes the scalar product in R3[54]. Geometrically this condition means that the projection of the angular velocity of the body to a fixed vector must equal zero. The equations of motion in the moving frame have the form I˙ Ω = IΩ×Ω + λγ, ˙γ=γ×Ω,(3.10) where Iis the inertia tensor of the rigid body, ×denotes the vector product in R3, and λ is a Lagrange multiplier chosen such that Ω(t) satisfies the above constraint, λ=−(IΩ×Ω,I−1γ) (I−1γ, γ).(3.11) The Veselova system (3.9), (3.10) is an LR system on the Lie group SO(3), which is the configuration space of the rigid body motion. After identification of Lie algebras (R3,×) and (so(3),[·,·]), the operator Iinduces the left-invariant metric ds2 I. The angular velocity correspond to Ω = g−1˙g, the velocity in the left trivialization T SO(3) ∼ =SO(3)×so(3). The vector fixed in the space corresponds to the right-invariant vector field γg=g·(g−1·a·g)∈ TgSO(3), a∈so(3), and the nonholonomic constraint (3.9) has the form hg−1·a·g, Ωi= 0. Once can check that the closed system (3.10), (3.11) has invariant measure with density (γ, I−1γ), as predicted by formula (3.6). Note that integrable potential perturbations of the Veselova system can be found in [54, 26]. Multidimensional Veselova problem. Now we proceed to a n-dimensional generalization of the Veselova system, describing the motion on the Lie group SO(n) with certain right-invariant nonholonomic constraints. 16 Let, as above, e1, . . . , enbe unit vectors that form a fixed orthogonal frame in the space Rn. Then, similarly to the generalized Suslov problem in Section 2, we define n-dimensional analog of (3.9) as follows: only infinitesimal rotations in the fixed 2-planes spanned by (e1, e2),...,(e1, en) are allowed. This implies the constraints hω, ei∧eji= 0,2≤i < j ≤n. (3.12) Equivalently, consider the right-invariant distribution Don T SO(n) whose restriction to the algebra so(n) is given by d= span{Ej∧Ek, k = 1, . . . , r, j = 1, . . . , n}, where Ei∧Ej form the basis in so(n). Since ei∧ej=g−1·Ei∧Ej·g, we have that constraints are ω∈ D =g−1·d·g= span{e1∧ei,2≤i≤n}. Remark 3.1. As for the multidimensional Suslov problem, the constraints (3.12) can be relaxed. However, in this case, the existence of the integrable LR system is still not known. That is why we keep using the above constraints (see Theorem 3.4). The LR system can be described by the Euler–Poincar´e equations (3.2, 3.3, 3.4) on the space so(n)×SO(n) with indefinite multipliers λpq, d dt (Iω) = [Iω, ω] + X 2≤p<q≤n λpq ep∧eq, ˙ei+ωei= 0, i = 1,...,n. (3.13) Here, as above, the components of e1,...,enplay the role of redundant coordinates on SO(n). Reduction. The orthogonal complement hof dis a Lie algebra, namely h= span{Ep∧Eq,2≤p < q ≤n}∼ =so(n−1). Therefore, the Veselova system can be treated as a generalized Chaplygin system on the principal bundle SO(n−1) −→ SO(n) ↓π Sn−1=SO(n−1)\SO(n) ,(3.14) where Sn−1is the n-dimensional sphere, realized as the unit sphere in Rn, Sn−1={q∈Rn−1, q2 1+···+q2 n= 1}, where we set q=e1. The moment map is then ω= Φ(q, ˙q) = q∧˙q. Thus, for solution e1(t), ω(t) = e1(t)∧˙e1(t) of (3.13), q(t) = e1(t) is a motion of a reduced system on the sphere Sn−1. The invariant measure. It appears that for some special inertia tensors, many of the calculations takes an especially simple form. Suppose that the operator Iis defined by a diagonal matrix A= diag(A1,...,An) in the following way I(Ei∧Ej) = AiAj det AEi∧Ej.(3.15) Notice that for n= 3 this corresponds to the well known three-dimensional vector formula I(x×y) = (det A)−1Ax ×Ay,A=I−1. 17 Under the condition (3.15) the reduced Lagrangian L(q, ˙q) and the right hand side of the Lagrange-d’Alambert equation (3.8) take the form L=1 2 det A[(A˙q, ˙q)(Aq, q)−(Aq, ˙q)2],(3.16) hIΦ(q, ˙q),prg−1hg[Φ(q, ˙q),Φ(q, ξ)]i=1 det AhAq ∧A˙q, prg−1hgξ∧˙qi =1 det A( ˙q, A ˙q)(Aq, ξ)−1 det A( ˙q, Aq)(A˙q, ξ) = Ψ(q, ˙q, ξ).(3.17) Here we used relation prg−1hgξ∧˙q=ξ∧˙qfor any admissible vector ξ= (ξ1,...,ξn)T∈ TqSn−1. Below we shall keep using the redundant coordinates qiand velocities ˙qi, in which the Lagrange equations have the form ∂L ∂qi−d dt ∂L ∂˙qi =πi+ Λqi, i = 1,...,n, (3.18) πi=∂Ψ ∂ξi =1 det A( ˙q, A ˙q)Aiqi−1 det A( ˙q, Aq)Ai˙qi, where Λ is a Lagrange multiplier. Now we want to represent the reduced LR system on T∗Sn−1as a restriction of a system on the Euclidean space R2n={q, p}. Note that L(q, ˙q) is degenerate in the redundant velocities ˙q, hence they cannot be expressed uniquely in terms of the redundant moments pi=∂L ∂˙qi≡1 det A(q, Aq)Ai˙qi−1 det A( ˙q, Aq)Aiqi.(3.19) In this case one can apply the Dirac formalism for Hamiltonian systems with constraints in the phase space (see, e.g., [18, 2, 43]). Namely, from (3.19) we find that (q, p) = 0, hence the cotangent bundle T∗Sn−1is realized as a subvariety of R2n= (q, p) defined by constraints φ1≡(q, q) = 1, φ2≡(q, p) = 0. Under these conditions, relations (3.19) can be uniquely inverted to yield ˙q=det A (q, Aq)A−1p−(p, A−1q)q.(3.20) On the other hand, we note that ∂L/∂qi=πi. Then, from (3.18) we obtain ˙p=−Λqand, from the condition ( ˙q, p) + (q, ˙p) = 0, ˙p=−Λq, Λ = det A(p, A−1p)−(p, q)(q, A−1p) (q, Aq).(3.21) The system (3.20), (3.21) on T∗Sn−1coincides with the restriction of the following system on R2n={q, p} ˙qi={qi,ˆ H}∗,˙pi={pi,ˆ H}∗−ˆπi, ˆπi(q, p) = πi(q, ˙q(q, p)),ˆ H=1 2det A(p, A−1p) (q, Aq), which is quasi-Hamiltonian with respect to the following Dirac bracket on R2n {F, G}∗={F, G}+{F, φ1}{G, φ2}−{F, φ2}{G, φ1} {φ1, φ2}, 18 {·,·} being the standard Poisson bracket on R2n. This system has explicit vector form ˙q=det A (q, Aq)A−1p−(p, A−1q) (q, q)q, ˙p=−det A(p, A−1p)(q, q)−(p, q)(q, A−1p) (q, Aq)(q, q)2q . (3.22) The bracket {·,·}∗is degenerate and possesses Casimir functions φ1, φ2specified above. Now, we can find the explicit form of the invariant measure of the reduced system. From (3.22) we find n X i=1 ∂˙qi ∂qi +∂˙pi ∂pi=−(n−2)det A(p, A−1q) (q, q) (q, Aq), which, in view of (3.19), takes the form (n−2)(q, A ˙q)/(q, Aq). Hence the extended system (3.22) possesses an invariant measure J= (Aq, q)−(n−2)/2dp1∧dq1∧···∧dpn∧dqn. Next, at points of T∗Sn−1, the standard volume form in R2ncan be represented as dp1∧dq1∧···∧dpn∧dqn=wn−1∧dΦ1∧dΦ2, where wis the restriction of the standard symplectic form dp1∧dq1+···+dpn∧dqnonto T∗Sn−1and Φ1,Φ2are certain functions of the Casimir functions φ1, φ2. Since the latter are invariants of the vector field V(p, q) given by (3.22), the Lie derivatives LVdΦ1,LVdΦ2 equal zero. Then, since LVJ= 0, we conclude that on T∗Sn−1, LV[(Aq, q)−(n−2)/2wn−1] = 0. As a result, we arrive at the following theorem. Theorem 3.3. The reduced LR system (3.20, 3.21) on T∗Sn−1possesses an invariant measure f(q) = (Aq, q)−(n−2)/2σ, σ =wn−1 where σis the canonical volume 2(n−1)-form on T∗Sn−1. Chaplygin reducing multiplier. As follows from Theorem 3.3, item 1) of Theorem 1.2, and the fact that the dimension of the reduced configuration manifold equals n−1, if our reduced LR system on T∗Sn−1were transformable to a Hamiltonian form by a time reparameterization, then the corresponding reducing multiplier Nshould be proportional to 1/p(q, Aq). Although Chaplygin’s reducibility theorem does not admit a straightforward multidimensional generalization, i.e., item 1) of Theorem 1.2 cannot be inverted, remarkably, for our reduced LR system on T∗Sn−1the inverse statement becomes applicable (see [26]). Theorem 3.4. 1). Under the time substitution dτ =pdet A/(Aq, q)dt and an appropriate change of momenta, the reduced LR system (3.18) or (3.20), (3.21) becomes a Hamiltonian system describing a geodesic flow on Sn−1with the Lagrangian L∗(q, dq/dτ) = 1 2(q, Aq)−1"Ad q dτ ,d q dτ (Aq, q)−Aq, d q dτ 2#.(3.23) 19 2). For A1< A2<···< Anthe latter system is algebraic completely integrable for any dimension n. In the spheroconic coordinates λ1,...,λn−1on Sn−1such that q2 i=(Ii−λ1)···(Ii−λn−1) Qj6=i(Ii−Ij), Ii=A−1 i(3.24) the Lagrangian L∗(q, dq/dτ)takes the St¨ackel form L∗=1 8 n−1 X k=1 Qs6=k(λk−λs) (λk−I1)···(λk−In)λkd dτ λk2 , and the evolution of λkis described by the Abel–Jacobi quadratures λk−1 1dλ1 2pR(λ1)+···+λk−1 n−1dλn−1 2pR(λn−1)=δk,n−1√2h dτ, (3.25) k= 1,···, n −1, where R(λ) = −(λ−I1)···(λ−In)λ(λ−c2)···(λ−cn−1),(3.26) h=L∗being the energy constant and c2,···, cn−1being other constants of motion (we set c1= 0). For generic values of these constants the corresponding invariant manifolds are (n−1)-dimensional tori. The item 1) of Theorem 3.4 is based on the relation between the reduced LR system to the celebrated Neumann system (see Theorem 3.5 below). Namely, consider the iso-energy submanifold Eh={L(q, ˙q) = h} ⊂ T Sn−1of the reduced Veselova system (3.18) and introduce another new time τ1by formula dτ1=sdet A2h (Aq, q)dt. (3.27) Theorem 3.5. Under the time substitution (3.27), the solutions q(t)of the reduced multidimensional Veselova system on Sn−1lying on the Ehtransforms to the solution of the integrable Neumann problem with the potential U(q) = 1 2(A−1q, q), d2 dτ2 1 q=−1 Aq+λq (3.28) corresponding to zero value of the integral F0=Adq dτ1 ,dq dτ1(Aq, q)−Aq, dq dτ12 −(Aq, q) (3.29) and vise versa. For n= 3, Theorem 3.5 is proved by Veselov and Veselova [55]. The proof for arbitrary dimensions is given in [26]. 3.3 Reconstructed Motion on D Now we consider the integrability of the original (unreduced) LR system on the rightinvariant distribution D⊂T SO(n), which is specified by constraints (3.12) and the leftinvariant metric given by (3.15). The relation between the reduced LR system and the 20 Neumann system described by Theorem 3.5 appears to be useful to reconstruct the motion on Dexactly. For this purpose we also shall make use of the correspondence between the Neumann system and the geodesic flow on a quadric (see Kn¨orrer [35]). Namely, consider a family of (n−1)-dimensional confocal quadrics in Rn, Q(α) = X2 1 α−A1 +···+X2 n α−An =−1, α ∈R.(3.30) Theorem 3.6. ([35]). Let X(s)be a geodesic on the quadric Q(0),sbeing a natural parameter. Then under the change of time ds =s(dX/ds, A−1dX/ds) (X, A−2X)dτ1(3.31) the unit normal vector q(τ1) = A−1X/|A−1X|is a solution to the Neumann system (3.28) corresponding to zero value of the integral (3.29) and vise versa. It is well known that the problem of geodesics on a quadric Q(0) is completely integrable, and qualitative behavior of the geodesics is described by the remarkable Chasles theorem (see e.g., [35, 43]): the tangent line ℓs={X(s) + σ dX/ds |σ∈R}of a geodesic X(s) on Q(0) is also tangent to a fixed set of confocal quadrics Q(α2),...,Q(αn−1)⊂Rn, where α2,...,αn−1are parameters playing the role of constants of motion (we set α1= 0). Now let nkbe the normal vector of the quadric Q(αk) at the touching point pk=ℓ∩Q(αk). Then another classical theorem of geometry says that the normal vectors n1,...,nn−1, together with the unit tangent vector γ=dX/ds, form an orthogonal basis in Rn. On the other hand, in [43], Moser made the following observation. Proposition 3.7. 1). Let xbe the position vector of a point on the line ℓs, which is tangent to geodesic X(s). Then in the new parameterization s1such that ds =−(X, A−2X)ds1 the evolution of the line is described by the Lax equations in n×nmatrix form d ds1L= [L,B],L= Πγ(A−x⊗x)Πγ,(3.32) B=A−1x⊗A−1γ−A−1γ⊗A−1x , (3.33) where Πγ=Id −(γ, γ)−1γ⊗γis the projection onto the orthogonal complement of γ in Rn. 2). The conserved eigenvalues of Lare given by the parameters α1= 0, α2,...,αn−1of the confocal quadrics and by an extra zero. The corresponding eigenvectors are parallel to the normal vectors n1=q,...,nn−1, and to γ. Now we are ready to describe generic solutions of the original LR system on D⊂T SO(n). Let q(τ1) be the solution of the Neumann system (3.28) with F0(q, q′) = 0, which is associated to a solution (q(t), p(t)) of the reduced LR system as described by Theorem 3.5. Let X= (q, Aq)−1/2Aq(s),n1=q(s),...,nn−1(s), γ(s) = dX ds (3.34) be the corresponding geodesic on Q(0) in the new parameterization sgiven by (3.31) and the unit eigenvectors of L. Also, according to (3.27) and (3.31) we can treat sas a known functions of the original time t. Then we have the following reconstruction theorem (see [26]). 21 Theorem 3.8. A solution (g(t),˙g(t)) of the original LR system on the distribution Dis given by the momentum map ω(t) = q∧˙qand the orthogonal frame formed by the unit vectors e1=q(t), e2=n2(t), . . . , en−1=nn−1(t), en=γ(t). The other solutions (g(t),˙g(t)) that are projected onto the same trajectory (q(t), p(t)) have the same ω, e1, while the rest of the frame is obtained by the orthogonal transformations, (e2(t)···en(t)) = (n2(t)···nn−1(t)γ(t)) R,(3.35) where the constant matrix Rranges over the group SO(n−1). Thus, from Theorems 3.8, 3.5 and the integrability properties of the Neumann system on T∗Sn−1we conclude that the phase space D⊂T SO(n) of the multidimensional Veselova LR system with the left-invariant metric defined by (3.15) is almost everywhere foliated by (n−1)-dimensional invariant tori, on which the motion is straight-line but not uniform. 3.4 Veselova Problem with Integrable Potentials and the Maupertuis Principle The Maupertuis principle. Consider a natural mechanical system on a compact Riemannian manifold (Q, ds2) with Hamiltonian h(q, p) = 1 2Pgij(q)pipj+v(q), where gij is the inverse of the metric tensor and v(q) is a smooth potential on Q. Let By the classical Maupertuis principle, the integral trajectories of the Hamiltonian vector field Xhwith h(q, p) = c > max v(q) coincide (up to a reparametrization) with the trajectories of another vector field XhJwith Hamiltonian hJ(q, p) = 1 2Xgij(q) c−V(q)pipj on the fixed iso-energy level Ec={h(q, p) = c}={hJ(q, p) = 1}. Namely, on Ecwe have dh = (c−v)dhJ(see [2]). The Hamiltonian flow of hJis the geodesic flow of the Jacobi metric ds2 J= (c−v(q))ds2,which is conformally equivalent to the original metric ds2. The Maupertuis principle can naturally be formulated for nonholonomic systems as well. Suppose the distribution Dis locally defined by ρ=n−kindependent 1-forms αi. Then the equations of the nonholonomic systems with Hamiltonians hand hJsubjected to the constraints ˙q∈Dqare given by ˙pi=−∂h ∂qi + ρ X i=1 λjαj(q)i,˙qi=∂h ∂pi , i = 1, . . . , n, (3.36) ˙pi=−∂hJ ∂qi + ρ X i=1 µjαj(q)i,˙qi=∂hJ ∂pi , i = 1,...,n. (3.37) On the iso-energy level Ec, the vector fields (3.36) and (3.37) are proportional and the Lagrange multipliers satisfy the relation λi=µi(c−v) (see [36]). One can verify that the construction goes through the Chaplygin reduction. This property can be used in producing non-trivial nonholonomic geodesic flows on SO(n) which, after the SO(n−1)-reduction, give rise to integrable systems on the sphere Sn−1. In the case of Hamiltonian systems, under a similar reduction, the Kovalevskaya and Goryachev–Chaplygin integrable cases of rigid body dynamics result in integrable geodesic flows on S2that possess additional polynomial integrals of degree 4 and 3 in momenta respectively (see [9]). 22 Veselova problem with potentials. Now let us go back to the n-dimensional Veselova problem and suppose that the n-dimensional rigid body is placed in an axisymmetric potential force field v=v(e1) (recall that {e1,...,en}are redundant coordinates on SO(n)). Then the equations of motion have the form d dt (Iω) = [Iω, ω] + ∂v ∂e1∧e1+X 2≤p<q≤n λpq ep∧eq, ˙ei+ωei= 0, i = 1,...,n, (3.38) togeteher with the constraints (3.12). The potential is SO(n−1)–invariant and induces a well defined reduced potential V(q) on the sphere Sn−1. Here V(q)≡v(e1)|e1=q. The perturbed reduced system with the inertia tensor (3.15) has the same Chaplygin reducing multiplier as the nonperturbed one. Therefore, in the new time τ, the reduced system becomes a natural mechanical system on the sphere with the kinetic energy (3.23) and the potential V(q). Let A1<···< An. It is known, that the most general separable potentials compatible with the metric (3.23) in the variables {λ1,...,λn−1}have the form V= n−1 X k=1 ∆k Qs6=k(λk−λs),(3.39) where ∆kare functions of the variable λkonly (see [33]). Note that this potentials are of the same form as the potentials compatible with the standard metric in the same coordinates (e.g., see [56]). Then, if ∆kis a Laurent polynomial in the variable λk, then the potential (3.39) is a Laurent polynomial in the coordinates variables q1,...,qn(see, e.g., [33, 19, 56]). In particular, the reduced Veselova problem with potential V(q) = α1(A−1q, q) + α2((A−1q, A−1q)−(A−1q, q)2) + n X i=1 αi+2 q2 i , αibeing arbitrary constants, is completely integrable. Now assume that v(e1) = α1(A−1e1, e1) + α2((A−1e1, A−1e1)−(A−1e1, e1)2) and that the total energy is bigger than maxSO(n)v. Let, as above, ds2 Ibe the left–invariant metric given by the inertia operator (3.15) and introduce the Jacobi metric ds2 J= (c−v(e1))ds2 I. From the above considerations and the Maupertuis principle we get the following result. Theorem 3.9. The SO(n−1)-reduction of the the nonholonomic geodesic flow of the metric ds2 Jwith the constraints (3.12) is completely integrable. The phase space T∗Sn−1is almost everywhere foliated by invariant (n−1)–dimensional Lagrangian tori with nonuniform quasi– periodic dynamics. The Lagrange case. In general, the operator (3.15) is not a physical inertia operator of a multidimensional rigid body. However, by taking A1=···=An−1,An> A1/2 we get Iω=Iω +ωI, I = diag(I1,...,I1, In), I1=A2 1 2 det A, In=A1An det A−A2 1 2 det A. In this case the system (3.38) represents the motion of a symmetric rigid body under the nonholonomic constraints. In the presence of the homogeneous gravitational force field in the direction e1we have v=Mg(C, e1), where gis the gravitational constant, Mis the mass and C= (C1,...,Cn) is the position of the center of mass of the body. If the mass center is placed on the 23 axis of the dynamical symmetry, then v=MgCne1nand the system (3.38) represents a multidimensional version of the Lagrange top (see [5]). In the new time τ, the reduced system is completely integrable according to a noncommutative version of the Liouvilee theorem. Appart from the Hamiltonian function, there are integrals arrising from the SO(n−1)–symmetry of the system, qi˜pj−qj˜pi,1≤i < j ≤n−1. As a result, the reduced phase space T∗Sn−1is foliated by two-dimensional invariant tori. Note that there is an another generalization of a heavy rigid body ([48]), which is based on the generalization of the three–dimensional Euler–Poisson equations to the Euler–Poisson equations on the semi-direct product so(n)×so(n). 4 L+R Systems 4.1 Definition and Invariant Measure of L+R Systems It appears that LR systems on a unimodular Lie group Gcan be viewed as a limit case of certain artificial systems on the same group, which also possess an invariant measure. The latter systems do not have a straightforward mechanical or geometric interpretation and arise as a “distortion” of a geodesic flow on Gwhose kinetic energy is given by a sum of a leftand right-invariant metrics. Geodesic flow on Gwith L+R metric. In addition to the nondegenerate linear operator Idefining the left-invariant metric (·,·)I, introduce a constant linear operator Γ0:g→g defining a right-invariant metric (·,·)Γon the n-dimensional compact Lie group G: for any vectors u, v ∈TgGwe put (u, v)Γ=hug−1,Γ0vg−1i. We take the sum of both metrics and consider the corresponding geodesic flow on Gdescribed by the Lagrangian l(ω, g) = 1 2hω, Iωi+1 2hgωg−1,Γ0gωg−1i ≡ 1 2hω, Iωi+hω, Γ(g)ωi, where Γ(g) = Adg−1Γ0Adgand Adgis regarded as a matrix operator acting on g. Suppose that the total inertia operator B(g) = I+ Γ(g) is nondegenerate and positive definite on the whole group G. The geodesic motion on the group is described by the Euler–Poincar´e equations ˙x= [x, ω] + g−1∂l ∂g , x =∂l ∂ω =Bω, (4.1) together with the kinematic equation ˙g=g·ω. In order to find explicit expression for g−1(∂l/∂g), we first note that for any Y∈g, hY, g−1(∂l/∂g)i=vY(l), where vYis the left-invariant vector field on Ggenerated by Y. Since the metric (·,·)Iis left-invariant, we have vY(l) = 1 2vY(hω, Γωi) = 1 2hω, Γad Yω+ ad T YΓωi=hΓω, [Y, ω]i=hY, ad ωΓωi. As a result, g−1(∂l/∂g) = ad ωΓω. Also, in view of the definition of Γ, its evolution is given by n×nmatrix equation ˙ Γ = Γad ω+ ad T ωΓ.(4.2) 24 Note that for compact group we have ad T ω=−ad ω, and ˙ Γ = [Γ,ad ω]. Equations (4.1), (4.2) form a closed system on the space g×Symm(n) with the coordinates ωi,Γij,i≤j= 1,...,n. Indeed, since Bis nondegenerate, the derivative ˙ωis uniquely defined from (4.1). L+R systems. Now we modify equations (4.1) by rejecting the term g−1(∂l/∂g). As a result, we obtain another system on the space g×Symm(n) d dt(Bω) = ad T ωBω, d dtΓ = Γ ad ω+ ad T ωΓ,B=I+ Γ.(4.3) This is generally non a Lagrangian system, and, in contrast to equations (4.1), (4.2), it possesses the “momentum” integral hBω, Bωi. In view of the structure of the kinetic energy, we shall refer to the system (4.3) as L+R system on G. Theorem 4.1. The L+R system (4.3) possesses the kinetic energy integral 1 2hω, Bωiand an invariant measure µ dω1∧···∧ωn∧dΓ11 ∧···∧Γnn with density µ=pdet(I+ Γ) .(4.4) Remark 4.1. The L+R systems can be also naturally considered on non-compact groups. Then Theorem 4.1 holds for unimodular groups as well. Recall that the group Gis unimodular if tr adω= 0. Proof of Theorem 4.1. First, replace d dt (Bω) with B˙ω+˙ Γω. Then, using (4.2) and the identity adωω= 0,we can represent equations (4.3) in the form B˙ω= ad T ωIω, ˙ Γ = Γad ω+ ad T ωΓ.(4.5) Using this form, we compute d dthω, Bωi= 2hω, B˙ωi+hω, ˙ Bωi = 2hω, ad T ωIωi+hω, Γad ωω+ ad T ωΓωi= 0. i.e., hω, Bωiis a first integral. Next, divergence ∆ of the phase flow of the system is calculated by the formula ∆ = n X i≤j ∂˙ Γij ∂Γij + n X i=1 ∂˙ωi ∂ωi .(4.6) In view of (4.2), the first sum equals Pn i≤j[(ad ω)jj + (ad ω)ii] = 0.Then we can write ∆ = tr (B−1U), Uij =∂(adT ωIω)i ∂ωj , i, j = 1, . . . , n. As follows from the first equation in (4.3), here we can put U= adIω+ adT ωI. In view of symmetry of B−1, the skew symmetric part of Udoes not contribute to the expression for ∆. The symmetric part of Uhas the form U+≡1 2(U+UT) = 1 2ad T ω(B−Γ) + (B −Γ) ad ω. 25 [34] Kharlamova-Zabelina, E I 1957 Rapid motion of a rigid body about a fixed point under the presence of a nonholonomic constraint, Vestnik Moskov. Univ., Ser. Mat. Mekh. Astr. Fiz. 12 no. 6, 25-34 (Russian). [35] Kn¨orrer H 1982 Geodesics on quadrics and a mechanical problem of C.Neumann. J. Reine Angew. Math. 334, 69–78 . [36] Koiller J 1992 Reduction of some classical non-holonomic systems with symmetry Arch. Rational Mech. 118 113-148. [37] Kolmogorov A N 1953 On dynamical systems with integral invariant on the torus. Dokl. Akad. Nauk SSSR 93 no. 5, 763-766. [38] Kozlov V V, Kolesnikov N N 1978 On theorems of dynamics Prikl. Mat. Mekh. 42 28–33. [39] Kozlov V V 1985 On the integrability theory of equations of nonholonomic mechanics. Advances in Mechanics,8, no.3, 85–107 (Russian); Regular and Chaotic Dynamics 7 (2002), no. 2, 161-176 [40] Kozlov V V 1987 On the existence of an invariant measure in smooth dynamical systems. Prikl. Mat. Meh. 51 no. 4, 538-545. [41] Kozlov V V 1988 Invariant measures of the Euler-Poincar´e equations on Lie algebras Funkt. Anal. Prilozh. 22 69-70 (Russian); English translation: 1988 Funct. Anal. Appl. 22 No.1, 58-59. [42] Marsden J E, Montgomery R, Ratiu T 1990 Reduction, symmetry and phases in mechanics. Memoirs of the American Mathematical Society, volume 88, number 436, Providence. [43] Moser J 1980 Various aspects of integrable Hamiltonian systems. In: Proc. CIME Conference. Bressanone, Italy, 1978. Prog. Math. 8, 233–290. [44] Moshchuk N K 1987 Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form. Prikl. Mat. Mekh. 51 no. 2, 223–229 (Russian) English translation in: J. Appl. Math. Mech. 51 (1987), no. 2, 172–177. [45] Neimark J I, Fufaev N A 1972 Dynamics of nonholonomic systems. Trans. of Math. Mon. 33, AMS Providence. [46] Neumann C 1859 De probleme quodam mechanico, quod ad primam integralium ultraellipticoram classem revocatum. J. Reine Angew. Math. 56. [47] Okuneva G G 1998 Integrable Variants of Non-Holonomic Rigid Body Problems Z. Angew. Math. Mech. 78 no. 12, 833-840. [48] Ratiu T 1982 Euler–Poisson equations on Lie algebras and the N–dimensional heavy rigid body Amer. J. Math. 104 409-448. [49] Schneider D. 2002 Nonholonomic Euler-Poincar´e Equations and Stability in Chaplygin’s Sphere. Dynamical Systems: An International Journal.,17 No. 2, 87–130 [50] 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. 32 [51] Strichartz R S 1986 Sub-Riemannian geometry J. Diff. Geometry 24 221-263; 1989 30 595-596. [52] Suslov, G.: Theoretical mechanic, Gostekhizdat, MoskvaLeningrad, 1951 (Russian). [53] Taimanov I A 1997 Integrable geodesic flows of nonholonomic metric, J. Dynam. Control Systems 3no.1, 129-147. [54] Veselov A P, Veselova L E 1986 Flows on Lie groups with nonholonomic constraint and integrable non–Hamiltonian systems Funkt. Anal. Prilozh. 20 no. 4, 65-66 (Russian); English translation: 1986 Funct. Anal. Appl. 20 no. 4, 308-309. [55] 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. [56] Wojciechowski S 1985 Integrable one-partical potentials related to the Neumann system and the Jacobi problem of geodesic motion on an ellipsoid, Phys. Lett. A 107 107-111. [57] Zenkov D V 1995 The Goemetry of the Routh Problem. J. Nonlin. Sci. 5, 503-519. [58] Zenkov D V, Bloch A M 2000 Dynamics of the n-dimensional Suslov problem. J. Geom. Phys. 34, no. 2, 121–136. [59] Zenkov D V, Bloch A M 2003 Invariant Measures of Nonholonomic Flows With Internal Degrees of Freedom. Nonlinearity 16, 1793–1807. [60] Zung, N. T. 2003 Torus actions and integrable systems, arXive: math.DS/0407455 33