There are two great post-Newtonian steps in classical mechanics. The first is the Lagrangian formulation and the accompanying principle of least action. The second is the Hamiltonian formulation, which is yet another way of writing Newtons equation of motion that uncovers what is really going on. This is where we start to see the deep and beautiful mathematical structure that underlies classical mechanics. It is also where we can make connections to what comes next, with quantum mechanics following very naturally from the Hamiltonian formulation.

We try to take some of the mystery out of the physicist’s manipulations of Classical Fields.

In this final chapter we introduce Hamilton--Jacobi theory along with its special insights into classical mechanics, and then go on to show how Erwin Schrödinger used the Hamilton--Jacobi equation to learn how to write his famous quantum-mechanical wave equation. In doing so, we will have introduced the reader to two of the ways classical mechanics served as a stepping stone to the world of quantum mechanics. Back in Chapter 5 we showed how Feynman’s sum-over-paths method is related to the principle of least action and the Lagrangian, and here we will show how Schrödinger used the Hamilton--Jacobi equations to invent wave mechanics. These two approaches, along with a third approach developed by Werner Heisenberg called “matrix mechanics,” turn out to be quantum-mechanical analogues of the classical mechanical theories of Newton, Lagrange, Hamilton, and Hamilton and Jacobi, in that they are describing the same thing in different ways, each with its own advantages and disadvantages.