A Pfaffian Generalization of Hamilton-Jacobi Theory

We study the motion of a submanifold under a prescribed smooth dynamical system by means of finite families of differential one-forms. The numbers of independent and dependent phase-space coordinates may differ. For forms involving only differentials of the independent coordinates, we derive an identity relating their transport equations to the motion of the submanifold. A matrix-rank condition ensures local recovery of all dependent coordinates from the form coefficients and equivalence of the evolution equations. We determine the least number of forms in this class and give explicit constructions using coordinates or first integrals. We also characterize when subtracting exact differentials makes a given family suitable for another projection while preserving recovery of all dependent coordinates. Scalar representations of the forms yield equations that include the classical Hamilton-Jacobi equation. A separate variational construction determines the full phase-space velocity by requiring each of several actions to be stationary. In the specified class, one action suffices in even dimension and two are necessary and sufficient in odd dimension. Linear and nonlinear five-dimensional examples illustrate both constructions.