On k-polycosymplectic Marsden-Weinstein reductions

TL;DR

This paper advances the theory of k-polysymplectic and k-polycosymplectic reduction, removing technical constraints, analyzing fibred manifolds, and applying results to physical systems like vibrating strings and membranes.

Contribution

It introduces a refined k-polycosymplectic Marsden--Weinstein reduction theory and explores its applications to Hamiltonian systems with symmetries, extending geometric reduction methods.

Findings

Improved reduction theory for k-polysymplectic manifolds.

Application to coupled vibrating strings and membranes.

Demonstration of k-polycosymplectic as a special case of k-polysymplectic geometry.

Abstract

We review and slightly improve the known k-polysymplectic Marsden--Weinstein reduction theory by removing some technical conditions on k-polysymplectic momentum maps by developing a theory of affine Lie group actions for k-polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k-polysymplectic manifolds, the so-called fibred ones, and we study their k-polysymplectic Marsden--Weinstein reductions. Previous results allow us to devise a k-polycosymplectic Marsden--Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k-polycosymplectic Hamiltonian systems with field symmetries. We show that k-polycosymplectic geometry can be understood as a particular type of k-polysymplectic geometry. Finally, a k-cosymplectic to l-cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k-cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given.