Variational aspect and kinetic theory of locally conformal dynamics

We present the locally conformal generalization of the Euler–Lagrange equations. We determine the dual space of the LCS Hamiltonian vector fields. Within this dual space, we formulate the Lie–Poisson equation that governs the kinetic motion of Hamiltonian systems in the context of local conformality. By expressing the Lie–Poisson dynamics in terms of density functions, we derive locally conformal Vlasov dynamics. In addition, we outline a geometric pathway that connects LCS Hamiltonian particle motion to locally conformal kinetic motion.