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

Krishan Rajaratnam ; Raymond G. McLenaghan ; Carlos Valero

Symmetry, integrability and geometry: methods and applications, Tome 12 (2016)

Résumé

We review the theory of orthogonal separation of variables of the Hamilton–Jacobi 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.

Détail

Keywords: completely integrable systems; concircular tensor; special conformal Killing tensor; Killing tensor; separation of variables; Stäckel systems; warped product; spaces of constant curvature; Hamilton–Jacobi equation; Schrödinger equation.

Bibliographie

[1] Benenti S., “Inertia tensors and Stäckel systems in the Euclidean spaces”, Rend. Sem. Mat. Univ. Politec. Torino, 50 (1992), 315–341 | MR | Zbl

[2] Benenti S., “Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation”, J. Math. Phys., 38 (1997), 6578–6602 | DOI | MR | Zbl

[3] Benenti S., “Special symmetric two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems”, Acta Appl. Math., 87 (2005), 33–91 | DOI | MR | Zbl

[4] Benenti S., “Separability in Riemannian manifolds”, SIGMA, 12 (2016), 013, 21 pp., arXiv: 1512.07833 | DOI | MR | Zbl

[5] Benenti S., Chanu C., Rastelli G., “The super-separability of the three-body inverse-square Calogero system”, J. Math. Phys., 41 (2000), 4654–4678 | DOI | MR | Zbl

[6] Calogero F., “Solution of a three-body problem in one dimension”, J. Math. Phys., 10 (1969), 2191–2196 | DOI

[7] Cariñena J. F., Rañada M. F., Santander M., Sanz-Gil T., “Separable potentials and a triality in two-dimensional spaces of constant curvature”, J. Nonlinear Math. Phys., 12 (2005), 230–252 | DOI | MR | Zbl

[8] Chanu C., Degiovanni L., McLenaghan R. G., “Geometrical classification of Killing tensors on bidimensional flat manifolds”, J. Math. Phys., 47 (2006), 073506, 20 pp., arXiv: math.DG/0512324 | DOI | MR | Zbl

[9] Cochran C. M., McLenaghan R. G., Smirnov R. G., “Equivalence problem for the orthogonal webs on the 3-sphere”, J. Math. Phys., 52 (2011), 053509, 22 pp., arXiv: 1009.4244 | DOI | MR | Zbl

[10] Crampin M., “Conformal {K}illing tensors with vanishing torsion and the separation of variables in the Hamilton–Jacobi equation”, Differential Geom. Appl., 18 (2003), 87–102 | DOI | MR | Zbl

[11] Crampin M., “On the orthogonal separation of variables in the Hamilton–Jacobi equation for geodesics in a Riemannian manifold”, Differential Geometry and its Applications, Matfyzpress, Prague, 2005, 453–466 | MR | Zbl

[12] Eisenhart L. P., “Separable systems of Stäckel”, Ann. of Math., 35 (1934), 284–305 | DOI | MR

[13] Haantjes J., “On $X_m$-forming sets of eigenvectors”, Indag. Math., 58 (1955), 158–162 | DOI | MR | Zbl

[14] Horwood J. T., “On the theory of algebraic invariants of vector spaces of Killing tensors”, J. Geom. Phys., 58 (2008), 487–501 | DOI | MR | Zbl

[15] Horwood J. T., McLenaghan R. G., “Transformation to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces”, J. Geom. Phys., 57 (2007), 1435–1440 | DOI | MR | Zbl

[16] Horwood J. T., McLenaghan R. G., “Orthogonal separation of variables for the Hamilton–Jacobi and wave equations in three-dimensional Minkowski space”, J. Math. Phys., 49 (2008), 023501, 48 pp. | DOI | MR | Zbl

[17] Horwood J. T., McLenaghan R. G., Smirnov R. G., “Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space”, Comm. Math. Phys., 259 (2005), 679–709, arXiv: math-ph/0605023 | DOI | MR | Zbl

[18] Horwood J. T., McLenaghan R. G., Smirnov R. G., “Hamilton–Jacobi theory in three-dimensional Minkowski space via Cartan geometry”, J. Math. Phys., 50 (2009), 053507, 41 pp. | DOI | MR | Zbl

[19] Kalnins E. G., “On the separation of variables for the Laplace equation $\Delta \Psi +K^{2}\Psi =0$ in two- and three-dimensional Minkowski space”, SIAM J. Math. Anal., 6 (1975), 340–374 | DOI | MR | Zbl

[20] Kalnins E. G., Separation of variables for Riemannian spaces of constant curvature, Pitman Monographs and Surveys in Pure and Applied Mathematics, 28, Longman Scientific Technical, Harlow; John Wiley Sons, Inc., New York, 1986 | MR | Zbl

[21] Kalnins E. G., Miller W. (Jr.), Separation of variables on $n$-dimensional Riemannian manifolds. II. The $n$-dimensional hyperboloid $\mathbf H_n$, University of Waikato Research Report, No 103, 1982

[22] Kalnins E. G., Miller W. (Jr.), “Separation of variables on $n$-dimensional Riemannian manifolds. I. The $n$-sphere $S_n$ and Euclidean $n$-space $R^n$”, J. Math. Phys., 27 (1986), 1721–1736 | DOI | MR | Zbl

[23] Kalnins E. G., Miller W. (Jr.), Reid G. J., “Separation of variables for complex Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for ${\rm S}_{nC}$ and ${\rm E}_{nC}$”, Proc. Roy. Soc. London Ser. A, 394 (1984), 183–206 | DOI | MR | Zbl

[24] Levi-Civita T., “Sulla integrazione della equazione di Hamilton–Jacobi per separazione di variabili”, Math. Ann., 59 (1904), 383–397 | DOI | MR | Zbl

[25] McLenaghan R. G., Smirnov R. G., “Intrinsic characterizations of orthogonal separability for natural Hamiltonians with scalar potentials on pseudo-Riemannian spaces”, J. Nonlinear Math. Phys., 9:1 (2002), 140–151 | DOI | MR

[26] McLenaghan R. G., Smirnov R. G., The D., “Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the $\rm O(4)$-symmetric Yang–Mills theories of Yatsun”, J. Math. Phys., 43 (2002), 1422–1440 | DOI | MR | Zbl

[27] McLenaghan R. G., Smirnov R. G., The D., “An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics”, J. Math. Phys., 45 (2004), 1079–1120 | DOI | MR | Zbl

[28] Morosi C., Tondo G., “Quasi-bi-Hamiltonian systems and separability”, J. Phys. A: Math. Gen., 30 (1997), 2799–2806, arXiv: solv-int/9702006 | DOI | MR | Zbl

[29] Nijenhuis A., “$X_{n-1}$-forming sets of eigenvectors”, Nederl. Akad. Wetensch. Proc. Ser. A., 54 (1951), 200–212 | DOI | MR | Zbl

[30] Nölker S., “Isometric immersions of warped products”, Differential Geom. Appl., 6 (1996), 1–30 | DOI | MR

[31] Olevskiĭ M. N., “Triorthogonal systems in spaces of constant curvature in which the equation $\Delta_2u+\lambda u=0$ allows a complete separation of variables”, Mat. Sb., 27 (1950), 379–426 | MR | Zbl

[32] O'Neill B., Semi-Riemannian geometry: with applications to relativity, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983 | MR | Zbl

[33] Petersen P., Riemannian geometry, Graduate Texts in Mathematics, 171, 2nd ed., Springer, New York, 2006 | DOI | MR | Zbl

[34] Rajaratnam K., Orthogonal separation of the Hamilton–Jacobi equation on spaces of constant curvature, Master's Thesis, University of Waterloo, 2014 http://hdl.handle.net/10012/8350 | MR

[35] Rajaratnam K., McLenaghan R. G., “Classification of Hamilton–Jacobi separation in orthogonal coordinates with diagonal curvature”, J. Math. Phys., 55 (2014), 083521, 16 pp., arXiv: 1404.2565 | DOI | MR | Zbl

[36] Rajaratnam K., McLenaghan R. G., “Killing tensors, warped products and the orthogonal separation of the Hamilton–Jacobi equation”, J. Math. Phys., 55 (2014), 013505, 27 pp., arXiv: 1404.3161 | DOI | MR | Zbl

[37] Rauch-Wojciechowski S., Waksjö C., “What an effective criterion of separability says about the Calogero type systems”, J. Nonlinear Math. Phys., 12:1 (2005), 535–547 | DOI | MR

[38] Schöbel K., “The variety of integrable Killing tensors on the 3-sphere”, SIGMA, 10 (2014), 080, 48 pp., arXiv: 1205.6227 | DOI | MR | Zbl

[39] Stäckel P., “Ueber die Bewegung eines Punktes in einer $n$-fachen Mannigfaltigkeit”, Math. Ann., 42 (1893), 537–563 | DOI | MR

[40] Thompson G., “Killing tensors in spaces of constant curvature”, J. Math. Phys., 27 (1986), 2693–2699 | DOI | MR | Zbl

[41] Waksjö C., Rauch-Wojciechowski S., “How to find separation coordinates for the Hamilton–Jacobi equation: a criterion of separability for natural Hamiltonian systems”, Math. Phys. Anal. Geom., 6 (2003), 301–348 | DOI | MR | Zbl