Factorization of the constants of motion

Abstract

A complete set of first integrals, or constants of motion, for a model system is constructed using “factorization”, as described below. The system is described by the effective Feynman Lagrangian L = [Formula: see text], with one of the simplest, nontrivial, potentials V(x) = (1/2)m ω2x2 selected for study. Four new, explicitly time-dependent, constants of the motion ci±, i = 1, 2 are defined for this system. While [Formula: see text]ci± ≠ 0, [Formula: see text]ci± = [Formula: see text]ci± + [Formula: see text]ci± + [Formula: see text]ci± + · · · = 0 along an extremal of L. The Hamiltonian H is shown to equal a sum of products of the ci±, and verifies [Formula: see text] = 0. A second, functionally independent constant of motion is also constructed as a sum of the quadratic products of ci±. It is shown that these derived constants of motion are in involution.PACS Nos.: 02.30.Jr, 02.30.Ik, 02.60.Cb, 02.30.Hq, 05.70.Ln, 02.50.–r