Cotangent bundle reduction and Routh reduction for polysymplectic manifolds

TL;DR

This paper extends reduction techniques to polysymplectic manifolds in field theories, connecting cotangent bundle and Routh reductions through the Routhian function, with practical examples illustrating the methods.

Contribution

It introduces a unified framework for cotangent bundle and Routh reduction in polysymplectic field theories, linking them via the Routhian and Legendre transformations.

Findings

Polysymplectic reduction theorems are applicable to invariant field theories.

A connection between cotangent bundle and Routh reduction is established.

Examples demonstrate the practical use of the reduction methods.

Abstract

We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. We end the paper with examples that illustrate the applicability of our results.