Group theoretical aspects of constants of motion and separable solutions in classical mechanics

AbstractThe aim of this paper is to establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only classical Lie point groups is insufficient. For this purpose the Lie-Bäcklund groups of tangent transformations, rigorously established by Ibragimov and Anderson, are used. It is also shown how these generalized groups induce Lie groups on Hamilton's equations. The generalization of the above results to any order partial differential equation, where the dependent variable does not appear explicitly, is obvious. In the second part of the paper we investigate a certain class of admissible operators of the time-independent Hamilton-Jacobi equation of any energy state including the zero state. It is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered