Canonical transformation to energy and ‘‘tempus’’ in classical mechanics

Abstract

In classical Hamiltonian dynamics for a system with a single degree of freedom a canonical transformation is made to new canonical variables in which the new canonical momentum is energy and its conjugate coordinate is called tempus. This canonical coordinate tempus conjugate to the energy is not necessarily the time t in which the system evolves, but is a function of the original generalized coordinate, the energy, and time t. For conservative systems tempus reduces to the time t, and the equations reduce to the Hamilton–Jacobi equation for Hamilton’s characteristic function. For periodic or almost periodic systems, the energy and tempus canonical variables act as a bridge to the action and angle canonical variables. Hamilton’s equations for the action and angle variables in the adiabatic limit involve a generalized Hannay (or geometrical) angle. A pendulum with a length varying in time is treated as an example.