Dynamical groups and spherical potentials in Classical Mechanics

Abstract

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation groupSO(3), i.e. according to the irreducible representationsD _j ofSO(3) (section 1).

The Lie algebras ofSO(4) andSO(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3).

In the last section, the Kepler problem is analyzed with the aid of theSO(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic toSO(4). Consequences with respect to the quantization problem are examined.