Reduction of exact symplectic manifolds and energy hypersurfaces

TL;DR

This paper develops two reduction methods for Hamiltonian systems on exact symplectic manifolds with Lie group symmetries, showing their equivalence and illustrating with examples.

Contribution

It introduces and proves the equivalence of two reduction schemes for Hamiltonian systems on exact symplectic manifolds with symmetries.

Findings

Two reduction schemes are shown to be equivalent.

A modified Marsden-Meyer-Weinstein reduction theorem is established.

Examples illustrate the reduction procedures.

Abstract

This article introduces two reduction schemes for Hamiltonian systems on an exact symplectic manifold admitting Lie group symmetries. It is demonstrated that these reduction procedures are equivalent by employing a modified Marsden-Meyer-Weinstein reduction theorem for exact symplectic manifolds and contact manifolds given by energy hypersurfaces. Each approach is illustrated through an example.