Killing tensors, Warped Products and The Orthogonal Separation of The Hamilton-Jacobi Equation
arXiv:1404.3161 · doi:10.1063/1.4861707
Abstract
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-Stackel 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.
References in corpus (1)
Cited by in corpus (5)
- Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
- Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature
- Orthogonal separation of variables for spaces of constant curvature
- Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional Hyperbolic and de Sitter spaces
- On a lower-dimensional Killing vector origin of irreducible Killing tensors