Non-Hamiltonian systems separable by Hamilton-Jacobi method

TL;DR

This paper demonstrates how to associate non-Hamiltonian bi-cofactor systems with classical Hamiltonian systems via metric deformation, preserving trajectories and integrability, and explores their separation properties and geodesic equivalences.

Contribution

It introduces a method to generate non-Hamiltonian systems from Hamiltonian ones using metric deformation, maintaining integrability and separability properties.

Findings

Bi-cofactor systems share separation curves with seed Hamiltonian systems.

Deformation of flat bi-cofactor systems can produce geodesically equivalent systems.

The approach preserves integrability and allows analysis of non-Hamiltonian dynamics.

Abstract

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.