Constructing a class of solutions for the Hamilton-Jacobi equation in field theory

TL;DR

This paper introduces a novel approach to formulate the Hamilton-Jacobi equation in field theory using jet-bundles and multi-symplectic manifolds, providing an algorithm for solutions based on boundary conditions.

Contribution

It develops a new geometric framework for the Hamilton-Jacobi equation in field theory and analyzes the limitations of the method related to boundary data compatibility.

Findings

Algorithm for associating solution classes to boundary conditions

Identification of intrinsic limits of the Hamilton-Jacobi method

Insights into boundary data compatibility constraints

Abstract

A new approach leading to the formulation of the Hamilton-Jacobi equation for field theories is investigated within the framework of jet-bundles and multi-symplectic manifolds. An algorithm associating classes of solutions to given sets of boundary conditions of the field equations is provided. The paper also puts into evidence the intrinsic limits of the Hamilton-Jacobi method as an algorithm to determine families of solutions of the field equations, showing how the choice of the boundary data is often limited by compatibility conditions.