Integrable geodesic flows of non-holonomic metrics

In the present article we show how to produce new examples of integrable dynamical systems of differential geometry origin. This is based on a construction of a canonical Hamiltonian structure for the geodesic flows of Carnot–Carathéodory metrics ([7, 17]) via the Pontryagin maximum principle. This Hamiltonian structure is achieved by introducing Lagrange multipliers bundles being the phase spaces of these Hamiltonian flows. These bundles are diffeomorphic to cotangent bundles but have another meaning. A transference to this phase space is given by a generalised Legendre transform. We analyse the geodesic flow of the left-invariant Carnot–Carathéodory metric on the three-dimensional Heisenberg group as a super-integrable Hamiltonian system (Theorem 1). Moreover, its super-integrability explains the foliation of its phase space into oneand two-dimensional invariant submanifolds as it was pointed out in [17]. The geodesic flows of left-invariant Carnot–Carathéodory metrics on Lie groups reduce to equations on Lie algebras in the same manner as the geodesic flows of left-invariant Riemannian metrics reduce to the Euler equations on Lie algebras ([1]) (Theorem 2). Most of these flows are integrable. Moreover, this reduction to equations on Lie algebras gives a Hamiltonian explanation for the description of such flows on three-dimensional Lie groups given in [17] by using Euler–Lagrange equations (the will to explain this in terms of integrability was the starting point for the present work). In comparison with the case of Riemannian geodesic flows there is another class of invariant flows, corresponding to left-invariant metrics and right-invariant distributions. In §6 we examine the simplest example of such flow on H and, in particular, show that this flow is integrable (Theorem 3). In §7 we consider an observation related to a Hamiltonian structure for the equations for the motion of a heavy rigid body with a fixed point. For completeness of explanation we discuss in §8 another approach to defining “straight lines” in non-holonomic geometry which does not arrive at Hamiltonian systems but some “straight line” flows have important mechanical meaning (for instance, the Chaplygin top ([6])). We also discuss some problems concerning dynamics and, in particular, integrability of these systems (see §9. Concluding remarks). The present article is dedicated to D. V. Anosov on his 60th birthday.