Entropy of Vlasov equilibria and Hamilton's principle

The following property is shown of the entropy functional constructed in previous work and associated with a Vlasov electrostatic or magnetostatic collective equilibrium (a static solution of the Vlasov equation): the vanishing of the first variation of the functional is equivalent to Hamilton's principle applied to a Lagrangian describing motion of the underlying system of particles compatible with the collective equilibrium, provided that the variations are associated with reversible processes. This property is shown in two cases: (i) a system of particles in Coulomb interaction admitting a collective (Vlasov) equilibrium in the presence of a scalar pressure; and (ii) a system of independent electrons in a background of fixed ions subject to an external electric field and to a magnetic field (created in part by electron currents) associated with a cylindrically or toroidally symmetric equilibrium in the presence of the electron-ion friction force and an anisotropic pressure.