Classical Canonical Transformations without the Use of Hamilton's Principle

A canonical transformation leading from a Hamiltonian system Σ to another Σ* is symmetrical in the sense that the transformation from Σ* to Σ is also canonical. By writing the transformation functions in such a way that the symmetry is exhibited explicitly, i.e., that neither Σ nor Σ* is singled out, one is naturally lead to consider an intermediate coordinate system S with 2n independent coordinates, n coordinates from Σ, and n from Σ*. Working from S it is then a straightforward matter to derive all the properties of canonical transformations. It is shown that the arbitrariness of the Lagrangian to within the addition of the total time derivative of an arbitrary function of the coordinates and the time follows from Lagrange's equations of motion. It is therefore also possible to derive the generating functions of canonical transformations without an appeal to Hamilton's principle.