Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

DOI

10.2991/jnmp.2005.12.2.5How to use a DOI?

Abstract

We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order sperintegrable if there are 2n - 1 (classically) functionally independent second-order symmetry operators. (The 2n - 1 is the maximum possible number of such symmtries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - E.G. Kalnins
AU  - J.M. Kress
AU  - W. Miller
PY  - 2005
DA  - 2005/05/01
TI  - Jacobi, Ellipsoidal Coordinates and Superintegrable Systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 209
EP  - 229
VL  - 12
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.2.5
DO  - 10.2991/jnmp.2005.12.2.5
ID  - Kalnins2005
ER  -