Lie Algebra Contractions and Separation of Variables on Two-Dimensional Hyperboloids. Coordinate Systems

TL;DR

This paper classifies all coordinate systems enabling separation of variables for the 2D Helmholtz equation on hyperboloids, explores their contractions to pseudo-euclidean and Euclidean planes, and uncovers new transition relations.

Contribution

It provides a comprehensive geometrical description of separable coordinate systems on hyperboloids and their contraction relations to flat geometries, including new transition findings.

Findings

Identified five orthogonal coordinate systems covering one-sheeted hyperboloids.

Established relations between hyperboloid and pseudo-euclidean plane coordinate systems.

Discovered previously unreported contraction transitions from hyperboloids to Euclidean plane.

Abstract

In this work the detailed geometrical description of all possible orthogonal and nonorthogonal systems of coordinates, which allow separation of variables of two-dimensional Helmholtz equation is given as for two-sheeted (upper sheet) $H_2$, either for one-sheeted ${\tilde H}_2$ hyperboloids. It was proven that only five types of orthogonal systems of coordinates, namely: pseudo-spherical, equidistant, horiciclic, elliptic-parabolic and elliptic system cover one-sheeted ${\tilde H}_2$ hyperboloid completely. For other systems on ${\tilde H}_2$ hyperboloid, well defined In\"on\"u--Wigner contraction into pseudo-euclidean plane $E_{1,1}$ does not exist. Nevertheless, we have found the relation between all nine orthogonal and three nonorthogonal separable systems of coordinates on the one-sheeted hyperboloid and eight orthogonal plus three nonorthogonal ones on pseudo-euclidean plane $E_{1,1}$. We could not identify the counterpart of parabolic coordinate of type II on $E_{1,1}$ among the nine separable coordinates on hyperboloid ${\tilde H}_2$, but we have defined one possible candidate having such a property in the contraction limit. In the light of contraction limit we have understood the origin of the existence of an additional invariant operator which does not correspond to any separation system of coordinates for the Helmholtz equation on pseudo-euclidean plane $E_{1,1}$. Finally we have reexamine all contraction limits from the nine separable systems on two-sheeted $H_2$ hyperboloid to Euclidean plane $E_2$ and found out some previously unreported transitions.