Canonical transformations: from the coordinate based approach to the geometric one

In this paper the theory of time-dependent and time-independent canonical transformations is considered from a geometric perspective. Both the geometric formalism and the coordinate based approach are described in detail. In particular, one-parameter groups of canonical transformations are geometrically identified with flows of Hamiltonian vector fields which, in turn, are their infinitesimal generators. Likewise, infinitesimal generators of invariance transformations are geometrically characterized. The main results are established in the form of theorems and the connection between the geometric and the coordinate based frameworks is remarked using concrete examples.