Are Hamiltonian Flows Geodesic Flows?

When a Hamiltonian system has a \Kinetic + Potential" structure, the resulting ow is locally a geodesic ow. But there may be singularities of the geodesic structure, so the local structure does not always imply that the ow is globally a geodesic ow. In order for a ow to be a geodesic ow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, double spherical pendulum, the Kovalevskaya top, and the N-body problem. We show that the ow of the reduced planar N-body problem and the reduced spatial 3-body are never geodesic ows except when the angular momentum is zero and the energy is positive. 1. Introduction Geodesic ows are always Hamiltonian. That is, given a Riemannian manifold M, the metric G can be considered as a function either on the tangent bundle or the cotangent bundle T M. The cotangent bundle has a natural symplectic structure and so the function G considered as a Hamiltonian deenes a Hamiltonian ow on T M. This ow is the same as the geodesic ow deened by the metric G when it is transfered to the cotangent bundle 1]. In this paper we begin the study of the inverse problem 10] and ask when is a Hamiltonian ow a geodesic ow, or more generally a reparameterization of a geodesic ow. There are important classical results along these lines.