Time-dependent Hamiltonian mechanics on a locally conformal symplectic manifold

TL;DR

This paper develops a geometric framework for time-dependent Hamiltonian dynamics on locally conformal symplectic manifolds, integrating contact geometry and providing applications to systems with time-dependent parameters.

Contribution

It introduces a generalized geometric Hamilton-Jacobi theory on lcs manifolds with time dependence, highlighting new geometric properties and applications.

Findings

Formulated a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds.

Presented examples of differential equations with time-dependent parameters.

Showed applications to physical systems with time-dependent features.

Abstract

In this paper we aim at presenting a concise but also comprehensive study of time-dependent (tdependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations and formulate a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers concerning locally conformal symplectic manifolds, here the introduction of the time dependency brings out interesting geometric properties, as it is the introduction of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out.