Non-Singular Canonical Variables.

The actions Ji defined by the phase integrals Ji = 1/2π ∮ pidqi may become singular. The simplest example is given by the actions of a Hamiltonian depending on the squares of the momenta. In this case, pi is proportional to $$ \dot q_i $$ and, as a consequence, the integral ⊂ pidqi is proportional to ⊂ $$ \int {\dot q_i^2 } dt $$ dt and, thus, sign definite. In other words, the integration path is always circulated in the same direction and the sign of Ji may not be reversed (Fig. 7.1). Consequently, the equations of motion in this variable are singular at Ji = 0. [ABSTRACT FROM AUTHOR]