Orthogonal Separation of The Hamilton-Jacobi Equation on Spaces of Constant Curvature
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Date
2014-04-24
Authors
Rajaratnam, Krishan
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
What is in common between the Kepler problem, a Hydrogen atom and a rotating black-
hole? These systems are described by different physical theories, but much information
about them can be obtained by separating an appropriate Hamilton-Jacobi equation. The
separation of variables of the Hamilton-Jacobi equation is an old but still powerful tool
for obtaining exact solutions.
The goal of this thesis is to present the theory and application of a certain type of
conformal Killing tensor (hereafter called concircular tensor) to the separation of variables
problem. The application is to spaces of constant curvature, with special attention to spaces
with Euclidean and Lorentzian signatures. The theory includes the general applicability of
concircular tensors to the separation of variables problem and the application of warped
products to studying Killing tensors in general and separable coordinates in particular.
Our first main result shows how to use these tensors to construct a special class of
separable coordinates (hereafter called Kalnins-Eisenhart-Miller (KEM) coordinates) on
a given space. Conversely, the second result generalizes the Kalnins-Miller classification
to show that any orthogonal separable coordinates in a space of constant curvature are
KEM coordinates. A closely related recursive algorithm is defined which allows one to
intrinsically (coordinate independently) search for KEM coordinates which separate a
given (natural) Hamilton-Jacobi equation. This algorithm is exhaustive in spaces of
constant curvature. Finally, sufficient details are worked out, so that one can apply these
procedures in spaces of constant curvature using only (linear) algebraic operations. As an
example, we apply the theory to study the separability of the Calogero-Moser system.
Description
Keywords
completely integrable systems, concircular tensor, special conformal Killing tensor, Killing tensor, separation of variables, Stackel systems, warped product, spaces of constant curvature, Hamilton-Jacobi equation, Schrodinger equation