The Lagrange\u2013Charpit Theory of the Hamilton\u2013Jacobi Problem

Abstract

The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle T^{*}Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.