ABSTRACT

The Hamiltonian procedure does not always offer any simplification to the concerned dynamical problem. But the almost symmetrical appearance of the coordinates and momenta in the guiding Hamilton’s equations facilitates development of formal theories such as the canonical transformations, Hamilton–Jacobi equation and action-angle variables. The purpose of this chapter is to understand how canonical transformations are constructed that offer considerable simplifications in writing down the equations of motion. The task is to transform the Hamiltonian to a new form through the introduction of a new set of coordinates and momenta such that with respect to the new set of variables the transformed Hamiltonian also leads to the form of canonical equations. The chapter deals briefly with the elements of a certain class of canonical variables called the action-angle variables defined over a bounded phase space that emerge from a complete separability of the Hamilton–Jacobi equation. Some interesting properties of such variables are explored.