The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].

The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].

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.

In the first two Chapters I outline the theory and background of separation of variables as an ansatz for solving fundamental partial differential equations (pdes) in Mathematical Physics. Two fundamental approaches will be highlighted, and more modern approaches discussed. In Chapter 3 I calculate the general trace-free conformal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determine the subcases with rotational symmetry and recover known examples pertaining to classical rotational coordinates. In Chapter 5 I obtain the induced action of the conformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of the conformal group to characterize, up to equivalence, the symmetric R-separable webs in E^3 that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Laplace equation.

In the first two Chapters I outline the theory and background of separation of variables as an ansatz for solving fundamental partial differential equations (pdes) in Mathematical Physics. Two fundamental approaches will be highlighted, and more modern approaches discussed. In Chapter 3 I calculate the general trace-free conformal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determine the subcases with rotational symmetry and recover known examples pertaining to classical rotational coordinates. In Chapter 5 I obtain the induced action of the conformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of the conformal group to characterize, up to equivalence, the symmetric R-separable webs in E^3 that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Laplace equation.