We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.
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January 2014
Research Article|
January 17 2014
Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation
Krishan Rajaratnam;
Krishan Rajaratnam
a)
Department of Applied Mathematics,
University of Waterloo
, Waterloo, Ontario N2L 3G1, Canada
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Raymond G. McLenaghan
Raymond G. McLenaghan
b)
Department of Applied Mathematics,
University of Waterloo
, Waterloo, Ontario N2L 3G1, Canada
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a)
Electronic mail: k2rajara@uwaterloo.ca
b)
Electronic mail: rgmclenaghan@uwaterloo.ca
J. Math. Phys. 55, 013505 (2014)
Article history
Received:
September 25 2013
Accepted:
December 26 2013
Citation
Krishan Rajaratnam, Raymond G. McLenaghan; Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation. J. Math. Phys. 1 January 2014; 55 (1): 013505. https://doi.org/10.1063/1.4861707
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