Master's defence | Krishan Rajaratnam, Orthogonal separation of the Hamilton-Jacobi equation on Spaces of Constant Curvature

Orthogonal separation of the Hamilton-Jacobi equation on Spaces of Constant Curvature

What is in common with 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.

In this presentation we will give an overview of the recently constructed geometric theory of orthogonal separable coordinates for the Hamilton-Jacobi equation on spaces of constant curvature. These coordinates are of interest in mathematical physics because they also allow for the separation of the Helmholtz equation in Euclidean space and the Klein-Gordon equation in Minkowski space.

The theory revolves around a certain type of conformal Killing tensor, hereafter called a concircular tensor. 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.

We will expose this theory by applying it to prove separability of well known coordinate systems in Euclidean 3-space, and then to study the separability properties of the Calogero-Moser system.