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

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.