Mechanical Hamiltonian systems with respect to linear Poisson structures and Jacobi-Reeb dynamics

TL;DR

This paper explores the connection between Jacobi-Reeb dynamics and mechanical Hamiltonian systems on vector bundles with linear Poisson structures, extending classical results to a broader geometric context.

Contribution

It introduces a framework linking Jacobi-Reeb dynamics with mechanical Hamiltonian systems using Jacobi bundle metrics, generalizing classical cotangent bundle results.

Findings

Established a relation between Jacobi-Reeb and Hamiltonian dynamics on vector bundles.

Extended classical Hamiltonian-Reeb dynamics correspondence to linear Poisson structures.

Provided geometric tools for analyzing mechanical systems with Jacobi structures.

Abstract

In this paper, we present a relation between Jacobi-Reeb dynamics and the dynamics associated with a mechanical Hamiltonian system with respect to a linear Poisson structure on a vector bundle. For this purpose, we will use the so-called Jacobi bundle metrics induced by the mechanical Hamiltonian system. These constructions extend classical results on the relation between standard mechanical Hamiltonian systems on cotangent bundles and Reeb dynamics.