Geodesics and Killing tensors in mechanics

Killing tensors give polynomial constants of the geodesic motion. The trajectories of a conservative mechanical system correspond to geodesics when the kinetic energy metric is conformally scaled to the Jacobi metric. Alternatively, the trajectories may be related to geodesics of some higher-dimensional warped product manifold. These two different ways of relating mechanical trajectories to geodesics are reviewed and compared. It is shown how a relation between Killing tensors on configuration space and the potential gives rise to Killing tensors on both the Jacobi and warped product manifolds