Poisson Geometry in Constrained Systems

TL;DR

This paper explores how Poisson geometry can be applied to constrained Hamiltonian systems, providing new insights into their structure and reduction methods, especially in relation to symplectic leaves and embeddings.

Contribution

It introduces conditions under which constrained systems can be represented as symplectic leaves within Poisson manifolds and relates Dirac brackets to Poisson structures.

Findings

Identification of symplectic dual pairs in constrained systems

Conditions for reduced phase space to be a symplectic leaf

Characterization of leafwise symplectic embeddings via cohomology

Abstract

Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.