On Non-autonomous Hamiltonian Dynamics, Dual Spaces, and Kinetic Lifts

TL;DR

This paper extends kinetic theories to time-dependent Hamiltonian systems using cosymplectic and cocontact geometries, linking them through momentum maps and analyzing various dynamical motions.

Contribution

It introduces geometric kinetic theories on cosymplectic and cocontact manifolds, expanding the framework to include time-dependent dynamics and hierarchical motion analysis.

Findings

Development of kinetic theories on cosymplectic and cocontact manifolds.

Linking of these theories via Poisson/momentum maps.

Hierarchical analysis of nine dynamical motions in cocontact geometry.

Abstract

Vlasov kinetic theory is the dynamics of a bunch of particles flowing according to symplectic Hamiltonian dynamics. More recently, this geometry has been extended to contact Hamiltonian dynamics. In this paper, we introduce geometric kinetic theories within the framework of cosymplectic and cocontact manifolds to extend the present literature to time-dependent dynamics. The cosymplectic and the cocontact kinetic theories are obtained in terms of both momentum variables and density functions. These alternative realizations are linked via Poisson/momentum maps. Furthermore, in cocontact geometry, we introduce a hierarchical analysis of nine distinct dynamical motions as various manifestations of Hamiltonian, evolution, and gradient flows.