Lie-algebra contractions and separation of variables. Three-dimensional sphere

Abstract

The Inonu-Wigner contraction from the SO(4) group to the Euclidean E(3) group is used to relate the separation of variables in Helmholtz equations for two corresponding homogeneous spaces. We show how the six systems of coordinates on the three-dimensional sphere contracted to nine systems of coordinates on Euclidean space. As a consequence of the Inonu-Wigner contraction we also consider contractions of the integrals of motion.