Abstract
On (pseudo)Riemannian manifolds of two and three dimensions, we list all metrics that admit a complete separation of variables in the Hamilton–Jacobi equation for geodesics. There are three different classes of separable metrics on two-dimensional surfaces. Three-dimensional manifolds admit six classes of separable metrics. Within each class, metrics are related by canonical transformations and a nondegenerate transformation of parameters that does not depend on coordinates.
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References
P. Havas, “Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation,” J. Math. Phys., 16, 1461–1468 (1975).
P. Havas, “Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation,” J. Math. Phys., 16, 2476–2489 (1975).
P. Stäckel, Über die Integration der Hamilton–Jacobischen Differentialgleichung mittelst Separation der Variablen, Habilitationsschrift, Halle (1891).
P. Stäckel, “Über die Bewegung eines Punktes in einer \(n\)-fachen Mannigfaltigkeit,” Math. Ann., 42, 537–563 (1893).
F. A. Dall’ Acqua, “Sulla integrazione delle equazioni di Hamilton–Jacobi per separazione di variabili,” Ann. Math., 66, 398–415 (1908).
P. Burgatti, “Determinazione dell’equazioni di Hamilton–Jacobi integrabili mediante la separazione delle variabili,” Rend. Accad. Lincei (Roma), 20, 108–111 (1911).
F. A. Dall’Acqua, “Le equazioni di Hamilton–Jacobi che si integrano per separazione di variabili,” Rend. Circ. Matem. Palermo, 33, 341–351 (1912).
M. S. Iarov-Iarovoi, “Integration of the Hamilton–Jacobi equation by the method of separation of variables,” J. Appl. Math. Mech., 27, 1499–1520 (1963).
F. Cantrijn, “Separation of variables in the Hamilton–Jacobi equation for non-conservative systems,” J. Phys. A, 10, 491–505 (1977).
T. Levi-Civita, “Sula integrazione della equazione di Hamilton–Jacobi per separazione di variabili,” Math. Ann., 59, 383–397 (1904).
L. P. Eisenhart, “Separable systems of Stäckel,” Ann. Math., 35, 284–305 (1934).
S. Benenti, “Separability in Riemannian manifolds,” SIGMA, 12, 013, 21 pp. (2016); arXiv: 1512.07833.
A. V. Bolsinov, A. Yu. Konyaev, and V. S. Matveev, “Orthogonal separation of variables for spaces of constant curvature,” arXiv: 2212.01605.
E. G. Kalnins and W. Miller, Jr., “Killing tensors and variable separation for Hamilton–Jacobi and Helmholtz equations,” SIAM J. Math. Anal., 11, 1011–1026 (1980).
E. G. Kalnins and W. Miller, Jr., “Killing tensors and nonorthogonal variable separation for Hamilton–Jacobi equations,” SIAM J. Math. Anal., 12, 617–629 (1981).
S. Benenti, “Separation of variables in the geodesic Hamilton–Jacobi equation,” in: Symplectic Geometry and Mathematical Physics (Aix-en-Provence, France, June 11–15, 1990, Progress in Mathematics, Vol. 99, P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman, eds.), Birkhäuser, Boston, MA (1991), pp. 1–36.
V. N. Shapovalov, “Stäckel spaces,” Siberian Math. J., 20, 790–800 (1979).
V. G. Bagrov and V. V. Obukhov, “Complete separation of variables in the free Hamilton–Jacobi equation,” Theoret. and Math. Phys., 97, 1275–1289 (1993).
V. V. Obukhov, The Stäckel Spaces in Gravity Theory [in Russian], Tomsk State Pedagogical University Press, Tomsk (2006).
V. V. Obukhov and K. E. Osetrin, Classification Problems in the Theory of Gravity [in Russian], Tomsk State Pedagogical University Press, Tomsk (2007).
B. Carter, “Hamilton–Jacobi and Schrödinger separable solutions of Einstein’s equations,” Commun. Math. Phys., 10, 280–310 (1968).
V. P. Frolov, P. Krtouš, and D. Kubizňák, “Black holes, hidden symmetries, and complete integrability,” Living Rev. Relativ., 20, 6, 221 pp. (2017); arXiv: 1705.05482.
V. V. Kozlov, “Quadratic conservation laws for equations of mathematical physics,” Russian Math. Surveys, 75, 445–494 (2020).
V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin, New York (1996).
M. O. Katanaev, “Complete separation of variables in the geodesic Hamilton–Jacobi equation,” arXiv: 2305.02222.
V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60), Springer, New York–Heidelberg (1978).
E. G. Kalnins and W. Miller, Jr., “Separable components for three-dimensional complex Riemennian spaces,” J. Differential Geom., 14, 221–236 (1979).
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This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 306–319 https://doi.org/10.4213/tmf10546.
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Katanaev, M.O. Separation of variables in the Hamilton–Jacobi equation for geodesics in two and three dimensions. Theor Math Phys 218, 264–275 (2024). https://doi.org/10.1134/S0040577924020065
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DOI: https://doi.org/10.1134/S0040577924020065