The generalized momenta of a dynamical system with n degrees of freedom may be given a Clebsch representation in terms of independent scalar functions and gradients thereof. The canonical equations imply certain differential relations which must be satisfied by these scalars. If the dynamical system is nonrelativistic, the differential relations are shown to be a generalization of the classical Hamilton–Jacobi theory. Similar results are obtained if the dynamical system is relativistic. A multiple integral variational principle, whose Euler–Lagrange equations imply the equations of motion and an equation of continuity, is formulated. As an example, it is shown that the Einstein field equations for an incoherent matter field and the geodesic equations of motion may be derived from a single fourfold variational principle. The usual energy-momentum tensor for such a matter field emerges as a by-product of the variational principle.