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