Existence of the Dynamic SymmetriesO4andSU3for All Classical Central Potential Problems

Abstract

It is found that all classical dynamic problems (relativistic as well as non-relativistic) involving central potentials, inherently possess both O4 and SU3 symmetry. This leads to a generalization of both the Runge-Lenz vector in the Kepler problem and the conserved symmetric tensor in the harmonic oscillator problem. For a general central potential, an explicit construction of the elements of the Lie algebra of O4 and SU3 in terms of canonical variables is given. The question of a possible quantum-mechanical analog is discussed. Also, a constructive technique is given for imbedding the Lorentz group and SU3 in an infinite-dimensional Lie algebra.