Lifted tensors and Hamilton–Jacobi separability

Abstract

Abstract. Starting from a bundle τ : E → R, the bundle π : J 1 τ * → E, which is the dual of the first jet bundle J 1 τ and a sub-bundle of T * E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1, 1) tensor R on E to both T * E and J 1 τ * . We discuss how an interplay between both lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat's conditions for separability of the time-dependent Hamilton-Jacobi equation in appropriate coordinates.