On the notion of phase in mechanics

The notion of phase plays an essential role in both semiclassical and quantum mechanics. But what is exactly a phase, and how does it change with time? It turns out that the most universal definition of a phase can be given in terms of Lagrangian manifolds by exploiting the properties of the Poincare-Cartan form. Such a phase is defined, not in configuration space, but rather in phase-space and is thus insensitive to the appearance of caustics. Surprisingly enough, this approach allows us to recover the Heisenberg-Weyl formalism without invoking commutation relations for observables