The globalization problem of the Hamilton-DeDonder-Weyl equations on a local $k$-symplectic framework

TL;DR

This paper addresses the challenge of globalizing Hamilton-DeDonder-Weyl equations within a local $k$-symplectic framework by introducing locally conformal $k$-symplectic manifolds and a global Lee one-form approach, enhancing the understanding of multi-Hamiltonian systems.

Contribution

It introduces the concept of locally conformal $k$-symplectic manifolds and develops a global Lee one-form method to extend local Hamiltonian properties to a global setting.

Findings

Defined locally conformal $k$-symplectic manifolds.

Developed a global Lee one-form approach.

Proposed a Hamilton--Jacobi equation in this framework.

Abstract

In this paper we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local $k$-symplectic framework and we introduce the notion of {\it locally conformal $k$-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton--Jacobi equation, which is specifically proposed in a l.c.k-s manifold.