De Donder-Weyl Equations and Multisymplectic Geometry

TL;DR

This paper explores the use of multisymplectic geometry to formulate De Donder-Weyl equations in classical field theories, highlighting differences from mechanics and discussing solution structures and Hamilton-Jacobi theory.

Contribution

It introduces a multisymplectic geometric framework for De Donder-Weyl equations, emphasizing solutions as integral manifolds and extending Hamilton-Jacobi theory to fields.

Findings

Solutions are integral manifolds of Hamiltonian multivector fields.

Solutions cannot be represented as points in the phase space.

Foliations of configuration space by solutions are discussed.

Abstract

Multisymplectic geometry is an adequate formalism to geometrically describe first order classical field theories. The De Donder-Weyl equations are treated in the framework of multisymplectic geometry, solutions are identified as integral manifolds of Hamiltonean multivectorfields. In contrast to mechanics, solutions cannot be described by points in the multisymplectic phase space. Foliations of the configuration space by solutions and a multisymplectic version of Hamilton-Jacobi theory are also discussed.