The Hamilton-Jacobi method for arbitrary rheo-linear dynamical systems

Abstract

The Hamilton-Jacobi method is briefly summarized and then applied to arbitrary rheo-linear systems with a single degree of freedom. Various means of finding a complete solution of the Hamilton-Jacobi equation and applying the Jacobi theorem to solve the canonical differential equations are discussed. Basic to the procedure is the separation of variables in the Hamilton-Jacobi equation which leads to a Riccati equation which must be solved for the particular rheonomic differential equation. Five different cases of complete solution are illustrated by a simple rheonomic example. As a direct application of the method, a number of canonical systems corresponding to many "named" equations are solved. They are the: Airy, Bessel, Chebyshev, Error function, Euler, Gegenbauer, Hermite, Hypergeometric, Kelvin, Kummer, Laguerre, Legendre, Jacobi, Mathieu, Spherical Bessel, Weber-Hermite and Whittaker equations. Finally, the conservation law is given for a forced and damped oscillator.