Are Hamiltonian flows geodesic flows?

Abstract

When a Hamiltonian system has a Kinetic + Potential structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, 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, the double spherical pendulum, the Kovalevskaya top, and the N-body problem. We show that the flow of the reduced planar N-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.