Conformal and Contact Kinetic Dynamics and Their Geometrization

We propose a conformal generalization of the reversible Vlasov equation of kinetic plasma dynamics, called conformal kinetic theory. To arrive at this formalism, we start with the conformal Hamiltonian dynamics of particles and lift this to the dynamical formulation of the associated kinetic theory. This theory represents a simple example of a geometric pathway from dissipative particle motion to dissipative kinetic motion. We also derive kinetic equations of a continuum of particles governed by the contact Hamiltonian dynamics, which can be interpreted in the context of relativistic mechanics. Once again we start with contact Hamiltonian dynamics and lift it to a kinetic theory, called contact kinetic dynamics. Finally, the contact kinetic theory projected to the conformal kinetic theory so that they form a geometric hierarchy.