The free energy of Maxwell–Vlasov equilibria

A previously derived expression [Phys. Rev. A 40, 3898 (1989)] for the energy of arbitrary perturbations about arbitrary Vlasov–Maxwell equilibria is transformed into a very compact form. The new form is also obtained by a canonical transformation method for solving Vlasov’s equation, which is based on Lie group theory. This method is simpler than the one used before and provides better physical insight. Finally, a procedure is presented for determining the existence of negative-energy modes. In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed.