  The perturbative variational formulation of the Vlasov-Maxwell equations is
presented up to third order in the perturbation analysis. From the second and
third-order Lagrangian densities, respectively, the first-order and
second-order Vlasov-Maxwell equations are expressed in gauge-invariant and
gauge-independent forms. Upon deriving the reduced second-order Vlasov-Maxwell
Lagrangian for the linear nonadiabatic gyrokinetic Vlasov-Maxwell equations,
the reduced Lagrangian densities for the linear drift-wave equation and the
linear hybrid kinetic-magnetohydrodynamic (MHD) equations are derived, with
their associated wave-action conservation laws obtained by Noether method. The
exact wave-action conservation law for the linear hybrid kinetic-MHD equations
is written explicitly. A new form of the third-order Vlasov-Maxwell Lagrangian
is derived in which ponderomotive effects play a crucial role.
