The Geometry of the Hamilton–Jacobi Equation

Abstract

A coisotropic submanifold of a cotangent bundle gives rise to several geometric objects that allow an appropriate and quite general discussion of the Hamilton–Jacobi equations. For example, the concept of “solution” appears to have two meanings: from a geometrical viewpoint, it is a Lagrangian submanifold of C (or, possibly, a Lagrangian set contained in C ), and, from an analytical viewpoint, it is a generating family satisfying a certain system of first–order PDE. One of the main problems related to a Hamilton–Jacobi equation is how to generate a (possibly unique) maximal solution from suitable initial conditions (Cauchy problem). We illustrate a geometrical construction of such a solution, by using the composition rule of symplectic relations, then we can transform this geometrical construction into an analytical method. Furthermore, other classical notions of geometrical optics, such as the system of rays and caustic of a system of rays, are more easily intelligible and manageable in a geometrical context.