Canonical Transformations

In this chapter, we consider in detail the canonical transformations, i.e. the possibly time-dependent change of variables which preserve the canonical structure of Hamilton’s equations. They are just families of symplectomorphisms of the phase space into itself parametrized by time. The main issues are to define the transformed Hamiltonian and to write the canonical transformation in an efficient way. This is accomplished using the generating functions: we discuss several versions of them, their mutual relations, and physical interpretation. We also present the theory of infinitesimal canonical transformations.