Hamilton-Jacobi Theory

Abstract

In this chapter, we present the classical Hamilton-Jacobi theory. This theory has played an enormous role in the development of theoretical and mathematical physics. On the one hand, it builds a bridge between classical mechanics and other branches of physics, in particular, optics. On the other hand, it yields a link between classical and quantum theory. We start with deriving the Hamilton-Jacobi equation and proving the classical Jacobi Theorem. We interpret the Hamilton-Jacobi equation geometrically as an equation for a Lagrangian submanifold of phase space which is contained in the coisotropic submanifold given by a level set of the Hamiltonian. Using this geometric picture, we extract a general method for solving initial value problems for arbitrary first order partial differential equations of the Hamilton-Jacobi type, the method of characteristics. It turns out that one can go beyond the case where a solution is generated by a single function on configuration space. To do so, one must consider Morse families, that is, families of Morse functions depending on additional parameters. In this chapter, we develop the theory of Morse families in a systematic way. Subsequently, we present the theory of critical points of Lagrangian submanifolds in cotangent bundles. This includes a topological characterization in terms of the Maslov class and a description of the topological data in terms of generating Morse families. Finally, we discuss applications in the spirit of geometric asymptotics. First, we study the eikonal equation of geometric optics, including the formation of caustics. Second, we analyse the transport equation and present a detailed study of its geometry. On this basis, we derive first order short wave asymptotic solutions for a class of first-order partial differential equations. In this analysis a key role is played by a consistency condition of topological type, the Bohr-Sommerfeld quantization condition. We discuss applications to the Helmholtz and to the Schrödinger equations.