Hamiltonian vector fields on almost symplectic manifolds

TL;DR

This paper studies Hamiltonian vector fields on almost symplectic manifolds, providing examples, new results, and a reduction theorem, expanding understanding of their geometric structure and applications.

Contribution

It introduces new examples of Hamiltonian vector fields on almost symplectic manifolds and establishes a reduction theorem of Marsden-Weinstein type.

Findings

Locally Hamiltonian vector fields generate a Dirac structure on the manifold.

A reduction theorem similar to Marsden-Weinstein is proven.

Examples of almost symplectic structures on tangent bundles are provided.

Abstract

Let $(M,\omega)$ be an almost symplectic manifold ($\omega$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_X\omega=0,d(i(X)\omega)=0$, and it is Hamiltonian if, furthermore, the 1-form $i(X)\omega$ is exact. Such vector fields were considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on $M$ and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.