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].
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On the Solution of the Hamilton-Jacobi Equation by the Method of Separation of Variables
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