Poisson brackets with prescribed Casimirs

TL;DR

This paper explores methods to construct Poisson brackets with specific Casimir functions on smooth manifolds, using almost symplectic and cosymplectic structures depending on the manifold's dimension, with various examples and applications.

Contribution

It introduces novel constructions of Poisson brackets with prescribed Casimirs on both even and odd-dimensional manifolds using almost symplectic and cosymplectic structures.

Findings

Construction of Poisson brackets on even-dimensional manifolds using almost symplectic structures.

Construction of Poisson brackets on odd-dimensional manifolds using almost cosymplectic structures.

Presentation of examples and applications demonstrating the methods.

Abstract

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.