Reduction of exact symplectic manifolds and energy hypersurfaces

TL;DR

The paper addresses reducing Hamiltonian dynamics on exact symplectic manifolds with Lie group symmetries by presenting two schemes: a modified Marsden–Meyer–Weinstein (MMW) reduction for exact symplectic manifolds and a compatible reduction on energy hypersurfaces via contact geometry. It proves the two procedures are equivalent under regular-value and quotientability assumptions, and it makes this precise through embeddings and a diffeomorphism between the reduced spaces. The authors construct explicit reduced objects $(P_{[\mu]},\theta_{[\mu]},\nabla_{[\mu]},h_{[\mu]})$, $(S_{[\mu]},\tilde{\eta}_{[\mu]})$, and $(M_{[\mu]},\eta_{[\mu]})$ and show the reduced flow corresponds to a Reeb-type vector field on the energy-hypersurface reductions. This work clarifies the interplay between symplectic and contact reductions in the presence of energy hypersurfaces and provides a concrete framework for symmetry reduction in exact Hamiltonian systems.

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.

Theorems & Definitions (19)

• Definition 1.1
• Definition 1.2
• Proposition 1.3
• Theorem 1.4
• Proposition 1.5
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2