Canonical Transformations and Hamilton—Jacobi Theory

Abstract

This chapter is divided into two parts: “Canonical Transformations” and “Hamilton—Jacobi Theory”. Concerning the subject of canonical transformations, we have seen in our study of cyclic coordinates that the integration of a dynamical system can generally be effected by transforming it into another dynamical system with fewer degrees of freedom by the use of cyclic coordinates. We also saw that, in the Hamiltonian formulation, the Hamiltonian does not contain the cyclic coordinates and the corresponding conjugate momenta are conserved. Such transformations that decrease the number of degrees of freedom of the system and leave Hamilton’s canonical equations of motion invariant, are called canonical transformations. We shall investigate these ideas in the context of the general theory that underlies the solutions of dynamical systems and is the basis for the Hamilton—Jacobi theory. This theory, to be discussed in Part II, gives the foundation for the modern theory of partial differential equations (PDEs) as applied to wave propagation phenomena.