On non-autonomous Hamiltonian dynamics, dual spaces, and kinetic lifts

Abstract

Vlasov kinetic dynamics fits within the Poisson framework, specifically in the Lie-Poisson form. In this context, each particle constituting the plasma follows classical symplectic Hamiltonian motion. More recently, this formulation has been extended to the kinetic formulation of a collection of particles flowing through contact Hamiltonian dynamics. In this paper, we introduce geometric kinetic theories within the frameworks of cosymplectic and cocontact manifolds, aiming to generalize the existing literature on symplectic kinetic theory and contact kinetic theory to include time-dependent dynamics. The cosymplectic and cocontact kinetic theories are formulated in terms of both momentum variables and density functions. These alternative realizations are connected through Poisson/momentum maps. Furthermore, in cocontact geometry, we present a hierarchical analysis of nine distinct dynamical motions, which serve as various manifestations of Hamiltonian, evolution, and gradient flows.