Variational structure of the Vlasov equation

The variational structure of the Vlasov–Maxwell integral equations is derived for a plasma equilibrium having two ignorable coordinates. It is shown that the kernel of the Maxwell equations is a self-adjoint integral operator. This operator may also be represented as a differential equation of arbitrary order. This representation is useful when the differential operator is truncated to finite order, yielding a system of intrinsically self-adjoint differential equations.