Existence of solutions for Hamiltonian field theories by the Hamilton-Jacobi technique

TL;DR

This paper proves the local existence of solutions to the Hamilton-Jacobi equation in field theory, enabling the derivation of general solutions of field equations from initial data using a specialized technique.

Contribution

It introduces a method to establish local solutions of the Hamilton-Jacobi equation in field theory tailored to initial data submanifolds.

Findings

Proved local existence of solutions for Hamilton-Jacobi equations in field theory.

Developed a technique to derive general solutions from initial data.

The method adapts to different initial submanifolds.

Abstract

The paper is devoted to prove the existence of a local solution of the Hamilton-Jacobi equation in field theory, whence the general solution of the field equations can be obtained. The solution is adapted to the choice of the submanifold where the initial data of the field equations are assigned. Finally, a technique to obtain the general solution of the field equations, starting from the given initial manifold, is deduced.