The Hamilton–Jacobi Theory

This chapter is dedicated to the Hamilton–Jacobi theory. After giving the historical motivations, we deduce the Hamilton–Jacobi equation from the theory of canonical transformations and prove Jacobi’s theorem. We present a number of applications to important systems. Then we describe how the Hamilton–Jacobi equations can be used to compute the geodesics on a Riemannian manifold and use this result to give the Hamilton–Jacobi description of the motion of a particle in a curved background in General Relativity. We introduce the technique of separation of variables and use it to solve several interesting systems, including the motion of a charged particle in a rotating charged Black Hole. Separable systems with configuration spaces \(\mathbb {R}^2\) and \(\mathbb {R}^3\) with flat Jacobi are classified in full detail. We introduce the notion of superseparability; as an example we construct all superseparable systems in \(\mathbb {R}^2\) . We discuss the wave interpretation of mechanics. In the final section we describe geometric optics á la Hamilton–Jacobi, deducing the eikonal equations and the Fermat principle from first physical principles (i.e. from the Maxwell equations for light waves).