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

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;Applied Mathematics

Rajaratnam, Krishan

University of Waterloo

关键词: 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;    Applied Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/8350/3/Rajaratnam_Krishan.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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 informationabout them can be obtained by separating an appropriate Hamilton-Jacobi equation. Theseparation of variables of the Hamilton-Jacobi equation is an old but still powerful toolfor obtaining exact solutions.The goal of this thesis is to present the theory and application of a certain type ofconformal Killing tensor (hereafter called concircular tensor) to the separation of variablesproblem. The application is to spaces of constant curvature, with special attention to spaceswith Euclidean and Lorentzian signatures. The theory includes the general applicability ofconcircular tensors to the separation of variables problem and the application of warpedproducts 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 ofseparable coordinates (hereafter called Kalnins-Eisenhart-Miller (KEM) coordinates) ona given space. Conversely, the second result generalizes the Kalnins-Miller classificationto show that any orthogonal separable coordinates in a space of constant curvature areKEM coordinates. A closely related recursive algorithm is defined which allows one tointrinsically (coordinate independently) search for KEM coordinates which separate agiven (natural) Hamilton-Jacobi equation. This algorithm is exhaustive in spaces ofconstant curvature. Finally, sufficient details are worked out, so that one can apply theseprocedures in spaces of constant curvature using only (linear) algebraic operations. As anexample, we apply the theory to study the separability of the Calogero-Moser system.