Hamilton-Jacobi theory in multisymplectic classical field theories

TL;DR

This paper extends the Hamilton-Jacobi theory within a multisymplectic geometric framework for classical field theories, unifying Lagrangian and Hamiltonian formalisms and providing new solution characterizations.

Contribution

It develops a geometric Hamilton-Jacobi framework for multisymplectic field theories, connecting solutions in both formalisms and including non-autonomous mechanical systems as special cases.

Findings

Unified Hamilton-Jacobi formalism for field theories

Characterization of solutions via distributions in jet bundles

Application to non-autonomous mechanical systems

Abstract

The geometric framework for the Hamilton-Jacobi theory developed in previous works is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.