Reduction of polysymplectic manifolds

TL;DR

This paper extends the Marsden-Weinstein reduction to polysymplectic manifolds, enabling the reduction of Hamiltonian systems with symmetries in classical field theories, and corrects previous inaccuracies in the literature.

Contribution

It introduces a generalized reduction procedure for polysymplectic manifolds, preserving their structure and providing new mathematical insights and applications.

Findings

Derived a polysymplectic reduction theorem analogous to Marsden-Weinstein

Established a polysymplectic version of the Kirillov-Kostant-Souriau theorem

Corrected previous errors in related mathematical literature

Abstract

The aim of this paper is to generalize the classical Marsden-Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which in- herit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogous to the Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed. Our procedure corrects some mistakes and inaccuracies in previous papers [29, 50] on this subject.