  We examine the development of the concept of parametric invariance in
classical mechanics, quantum mechanics, statistical mechanics, and
thermodynamics, and particularly its relation to entropy. The parametric
invariance was used by Ehrenfest as a principle related to the quantization
rules of the old quantum mechanics. It was also considered by Rayleigh in the
determination of pressure caused by vibration, and the general approach we
follow here is based on his. Specific calculation of invariants in classical
and quantum mechanics are determined. The Hertz invariant, which is a volume in
phase space, is extended to the case of a variable number of particles. We show
that the slow parametric change leads to the adiabatic process, allowing the
definition of entropy as a parametric invariance.
