The aim of this chapter is to consider various formalisms of classical dynamics and their equivalence or lack of it. These considerations are very important when studying the ‘classical limit’ of quantum mechanics, i.e. which particular formulation of quantum mechanics will give rise, in the limiting process, to a particular formulation of classical mechanics?

Our review of basic concepts and tools in classical mechanics begins with the definition of Poisson brackets on functions on a manifold. The Poisson bracket is any map which is antisymmetric, bilinear, satisfies the Jacobi identity and obeys a fourth property (derivation) that relates the Poisson bracket with the commutative associative product. Symplectic geometry is then outlined, and an intrinsic definition of the Poisson bracket is given within that framework. The maps which preserve the Poisson-bracket structure are canonical transformations. They are presented in an implicit form in terms of the generating functions. The expressions for the new canonical variables in terms of the old canonical variables are non-linear in general, and they can be made explicit only locally. In the case of linear canonical transformations it can be extended to global definitions. Once a symplectic potential is selected one may identify four classes of generating functions of canonical transformations, and they are all presented. This makes it possible to cast the equations of motion in the simplest possible form after performing one set of canonical transformations. The problem of solving the Hamilton equations is then replaced by the analysis of a partial differential equation known as the Hamilton–Jacobi equation.