Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems

TL;DR

This paper introduces mechanical presymplectic structures as a more suitable framework than cosymplectic geometry for reducing symmetric time-dependent Hamiltonian systems, extending Marsden-Weinstein reduction to this new setting.

Contribution

The authors develop a Marsden-Weinstein reduction theory for mechanical presymplectic structures, overcoming limitations of previous cosymplectic approaches in time-dependent Hamiltonian systems.

Findings

Mechanical presymplectic structures generalize cosymplectic structures.

The new reduction method applies to systems where Albert's approach fails.

Examples demonstrate the effectiveness of the new reduction technique.

Abstract

In 1986, Albert proposed a Marsden-Weinstein reduction process for cosymplectic structures. In this paper, we present the limitations of this theory in the application of the reduction of symmetric time-dependent Hamiltonian systems. As a consequence, we conclude that cosymplectic geometry is not appropriate for this reduction. Motived for this fact, we replace cosymplectic structures by more general structures: mechanical presymplectic structures. Then, we develop Marsden-Weinstein reduction for this kind of structures and we apply this theory to interesting examples of time-dependent Hamiltonian systems for which Albert's reduction method doesn't work.