Feynman propagator for time-dependent Lagrangians possessing an invariant quadratic in momentum

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown that the propagator for such a time-dependent system is related to the propagator of an associated time-dependent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of previous workers is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach. (author)