Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature

Rajaratnam, Krishan; McLenaghan, R. G.

doi:10.1063/1.4893335

Cited by 10 publications

(26 citation statements)

References 9 publications

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“…Bpζq " ζ 3 pζ´aq, B U K pζq " pζ´aq, ppζq " pζ´uqpζ´vqpζ´wq where u, v, w are the eigenfunctions of L. Application of equations (13) and (14) yield the transformation equations between pu, v, wq and pη, y, ξ, zq on H 3 and dS 3 , as appropriate. As usual, we write out the transformation equations below in terms of pseudo-Cartesian coordinates pt, x, y, zq associated with our null Cartesian coordinates pη, y, ξ, zq.…”

Section: Restriction To Hmentioning

confidence: 99%

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Valero

McLenaghan

2019

Journal of Mathematical Physics

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We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for Hyperbolic 3-space agree with those in the literature, while the results for de Sitter 3-space appear to be new.

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Section: Restriction To Hmentioning

confidence: 99%

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Valero

McLenaghan

2019

Journal of Mathematical Physics

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“…For simplicity we assume K has a single multidimensional eigenspace D. Then as in Section 4.2, the pair (D ⊥ , D) induces a warped product B × ρ F which is locally isometric to M . One can show that both B and F are spaces of constant curvature with B having the same curvature as M [35,Section 4].Then by Proposition 4.7, K induces a Killing tensor K F on F , by restriction. One can then recursively construct the KEM web by applying the above theorem to the ChKT K F on the space of constant curvature F , and then analyzing the resulting concircular tensor as above.…”

Section: Necessity Of Kem Webs In Spaces Of Constant Curvaturementioning

confidence: 99%

“…10 (separable webs in spaces of constant curvature[35]). In a space of constant curvature, every orthogonal separable web is a KEM web.…”

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confidence: 99%

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

Rajaratnam¹,

McLenaghan²,

Valero³

2016

SIGMA

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Abstract. We review the theory of orthogonal separation of variables of the HamiltonJacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.

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“…Then there is a non-trivial concircular tensor L defined on M such that each eigenspace of K is L-invariant, i.e. L is diagonalized in coordinates adapted to the eigenspaces of K. ♦ A rigorous proof will be given in [RM14a]. In Riemannian spaces of constant curvature, this theorem can be proven by connecting the classification of separable metrics given by Kalnins and Miller in [Kal86] with Theorem 6.1.…”

mentioning

confidence: 99%

“…For a space of constant curvature with arbitrary signature, it can be shown that the classification given by Kalnins and Miller can be generalized in such a way that the separable metrics still satisfy the hypothesis of Theorem 6.1. This generalization will be given in [RM14a].…”

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confidence: 99%

Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation

Rajaratnam

McLenaghan

2014

Journal of Mathematical Physics

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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. C 2014 AIP Publishing LLC. [http://dx.

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Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature

Cited by 10 publications

References 9 publications

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

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

Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation

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