Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds

TL;DR

This paper develops a Hamilton-Jacobi formalism tailored for locally conformally symplectic manifolds, enabling better integration of dynamical systems in physical theories with line bundle Hamiltonians.

Contribution

It introduces a local and global Hamilton-Jacobi framework on locally conformally symplectic manifolds, connecting local behavior with global structure using the Lichnerowicz-deRham differential.

Findings

Established local Hamilton-Jacobi equations on l.c.s. manifolds

Connected local and global descriptions of the theory

Demonstrated how local solutions can be glued for global understanding

Abstract

In this article we provide a Hamilton-Jacobi formalism in locally conformally symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the suitability of this theory as an integration method for dynamical systems, whilst our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a local l.c.s. Hamilton-Jacobi in subsets of the general manifold, and then provide a global view by using the Lichnerowicz-deRham differential. We show a comparison between the global and local description of a l.c.s. Hamilton--Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton-Jacobi theory.