Lifted tensors and Hamilton-Jacobi separability

TL;DR

This paper explores the geometric structures underlying time-dependent Hamiltonian systems, showing how lifted tensors and their integrability conditions relate to the separability of the Hamilton-Jacobi equation.

Contribution

It establishes a geometric framework linking lifted tensors and distributions to Forbat's separability conditions in time-dependent Hamiltonian systems.

Findings

Lifted tensors induce distributions whose integrability relates to separability.

The geometric conditions derived match Forbat's criteria for Hamilton-Jacobi separability.

Provides a coordinate-free approach to understanding Hamilton-Jacobi separability.

Abstract

Starting from a bundle E over R, the dual of the first jet bundle, which is a co-dimension 1 sub-bundle of the cotangent bundle of 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 of these manifolds. We discuss how an interplay between these 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.