Solutions of Vlasov's equation are given in terms of the Hamilton-Jacobi function S for the characteristics. Such solutions are appropriate in order to derive equations for macroscopic quantities which are of interest, for instance, in turbulence theories. In this context it is important that the Hamilton-Jacobi-theory leads to a simple straightforward nonsecular perturbation theory. The method is applied to a one-dimensional plasma which is homogeneous in the unperturbed state.