Hamilton-Jacobi theory in Cauchy data space

TL;DR

This paper extends Hamilton-Jacobi theory to the Cauchy data space in classical field theories, connecting it with the multisymplectic geometric approach to provide a unified framework.

Contribution

It develops a Hamilton-Jacobi theory in the Cauchy data space and relates it to the existing multisymplectic geometric formulation.

Findings

Established Hamilton-Jacobi theory in the Cauchy data space.

Connected the Cauchy data space approach with multisymplectic geometry.

Provided a unified geometric framework for classical field theories.

Abstract

Recently, M. de Le\'on el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.