Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 117, 30 pages      arXiv:1607.00712      https://doi.org/10.3842/SIGMA.2016.117
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

Krishan Rajaratnam a, Raymond G. McLenaghan b and Carlos Valero b
a) Department of Mathematics, University of Toronto, Canada
b) Department of Applied Mathematics, University of Waterloo, Canada

Received September 30, 2015, in final form December 12, 2016; Published online December 21, 2016

Abstract
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.

Key words: completely integrable systems; concircular tensor; special conformal Killing tensor; Killing tensor; separation of variables; Stäckel systems; warped product; spaces of constant curvature; Hamilton-Jacobi equation; Schrödinger equation.

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