6 - TOPICS IN HAMILTONIAN DYNAMICS

Summary

CHAPTER OVERVIEW

Most dynamical systems cannot be integrated in closed form: the equations of motion do not lend themselves to explicit analytic solution. There exist strategies, however, for approximating solutions analytically and for obtaining qualitative information about even extremely complex and difficult dynamical systems. Part of this chapter is devoted to such strategies for Hamiltonian dynamics.

The first two sections describe Hamilton–Jacobi (HJ) theory and introduce action–angle (AA) variables on T^*ℚ. HJ theory presents a new and powerful way for integrating Hamilton's canonical equations, and motion on T^*ℚ is particularly elegant and easy to visualize when it can be viewed in terms of AA variables. Both schemes are important in their own right, but they also set the stage for Section 6.3.

THE HAMILTON–JACOBI METHOD

It was mentioned at the end of Worked Example 5.5 that there would be some advantage to finding local canonical coordinates on T^*ℚ in which the new Hamiltonian function vanishes identically, or more generally is identically equal to a constant. In this section we show how to do this by obtaining a Type 1 generating function for a CT from the initial local coordinates on T^*ℚ to new ones, all of which are constants of the motion. The desired generating function is the solution of a nonlinear partial differential equation known as the Hamilton–Jacobi (or HJ) equation. Unfortunately, the HJ equation is in general not easy to solve.