Symmetry and Degeneracy in Classical Mechanics

Abstract

Recently it has been shown by several authors that both an $O\left(4\right)\mathrm{}$ and an $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry, heretofore associated with the nonrelativistic Kepler and three-dimensional isotropic harmonic-oscillator problems, respectively, are automatically possessed by all classical central potentials to the extent that the Poisson bracket forms of the Lie alegbras of these groups can be explicitly constructed. This result has been extended by Mukunda to be a property of all classical dynamical problems involving three degrees of freedom independent of the functional form of the Hamiltonian. We investigate the interrelations among the classical mechanical degeneracy, the simply periodic nature of motions, and the separability of Hamilton-Jacobi equations, and the question to what extent the invariance of the Hamiltonian of a classical system under the Poisson bracket forms of the Lie algebras constitutes a higher symmetry in the global and dynamical sense. We show that for a large class of classical systems the occurrence of degeneracies is a direct consequence of the separability of the Hamilton-Jacobi equations in a continuous family of coordinate systems and that, as such, Lie algebras do not by themselves automatically constitute higher symmetries unless a finitely multivalued realization of the corresponding group in the phase space of the system exists.