Separation of variables in the Hamilton–Jacobi equation for geodesics in two and three dimensions

M. O. Katanaev

Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 2, pp. 306-319

Résumé

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.

Détail

Keywords: Hamilton–Jacobi equation, separation of variables, geodesic.

Bibliographie

[1] P. Havas, “Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation”, J. Math. Phys., 16:7 (1975), 1461–1468 | DOI | MR

[2] P. Havas, “Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation”, J. Math. Phys., 16:12 (1975), 2476–2489 | DOI | MR

[3] P. Stäckel, Über die Integration der Hamilton–Jacobischen Differentialgleichung mittelst Separation der Variablen, Habilitationsschrift, Halle, 1891

[4] P. Stäckel, “Über die Bewegung eines Punktes in einer $n$-fachen Mannigfaltigkeit”, Math. Ann., 42:4 (1893), 537–563 | DOI | MR

[5] F. A. Dall' Acqua, “Sulla integrazione delle equazioni di Hamilton–Jacobi per separazione di variabili”, Ann. Math., 66:3 (1908), 398–415 | DOI | MR

[6] P. Burgatti, “Determinazione dell'equazioni di Hamilton–Jacobi integrabili mediante la separazione delle variabili”, Rend. Accad. Lincei (Roma), 20:1 (1911), 108–111 | Zbl

[7] F. A. Dall'Acqua, “Le equazioni di Hamilton–Jacobi che si integrano per separazione di variabili”, Rend. Circ. Matem. Palermo, 33:1 (1912), 341–351 | DOI

[8] M. S. Yarov-Yarovoi, “Ob integrirovanii uravneniya Gamiltona–Yakobi metodom razdeleniya peremennykh”, PMM, 27:6 (1963), 973–987 | DOI | MR

[9] F. Cantrijn, “Separation of variables in the Hamilton–Jacobi equation for non-conservative systems”, J. Phys. A, 10:4 (1977), 491–505 | DOI | MR

[10] T. Levi-Civita, “Sula integrazione della equazione di Hamilton–Jacobi per separazione di variabili”, Math. Ann., 59:3 (1904), 383–397 | MR

[11] L. P. Eisenhart, “Separable systems of Stäckel”, Ann. Math., 35:2 (1934), 284–305 | DOI | MR

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

[13] A. V. Bolsinov, A. Yu. Konyaev, V. S. Matveev, Orthogonal separation of variables for spaces of constant curvature, arXiv: 2212.01605

[14] E. G. Kalnins, W. Miller, Jr., “Killing tensors and variable separation for Hamilton–Jacobi and Helmholtz equations”, SIAM J. Math. Anal., 11:6 (1980), 1011–1026 | DOI | MR

[15] E. G. Kalnins, W. Miller, Jr., “Killing tensors and nonorthogonal variable separation for Hamilton–Jacobi equations”, SIAM J. Math. Anal., 12:4 (1981), 617–629 | DOI | MR

[16] S. Benenti, “Separation of variables in the geodesic Hamilton–Jacobi equation”, Symplectic Geometry and Mathematical Physics (Aix-en-Provence, France, June 11–15, 1990), Progress in Mathematics, 99, eds. P. Donato, C. Duval, J. Elhadad, G. M. Tuynman, Birkhäuser, Boston, MA, 1991, 1–36 | MR

[17] V. N. Shapovalov, “Prostranstva Shtekkelya”, Sib. matem. zhurn., 20:5 (1979), 1117–1130 | DOI | MR

[18] V. G. Bagrov, V. V. Obukhov, “Polnoe razdelenie peremennykh v svobodnom uravnenii Gamiltona–Yakobi”, TMF, 97:2 (1993), 250–269 | DOI | MR | Zbl

[19] V. V. Obukhov, Shtekkelevy prostranstva v teorii gravitatsii, Izd-vo Tom. gos. ped. un-ta, Tomsk, 2006

[20] V. V. Obukhov, K. E. Osetrin, Klassifikatsionnye problemy v teorii gravitatsii, Izd-vo Tom. gos. ped. un-ta, Tomsk, 2007

[21] B. Carter, “Hamilton–Jacobi and Schrödinger separable solutions of Einstein's equations”, Commun. Math. Phys., 10:4 (1968), 280–310 | DOI | MR

[22] V. P. Frolov, P. Krtouš, D. Kubizňák, “Black holes, hidden symmetries, and complete integrability”, Living Rev. Relativ., 20 (2017), 6, 221 pp., arXiv: 1705.05482 | DOI

[23] V. V. Kozlov, “Kvadratichnye zakony sokhraneniya uravnenii matematicheskoi fiziki”, UMN, 75:3 (2020), 55–106 | DOI | DOI | MR | Zbl

[24] V. V. Kozlov, Simmetrii, topologiya i rezonansy v gamiltonovoi mekhanike, Izd-vo Udmurt. un-ta, Izhevsk, 1995 | MR

[25] M. O. Katanaev, Complete separation of variables in the geodesic Hamilton–Jacobi equation, arXiv: 2305.02222

[26] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1989 | DOI | MR | MR | Zbl

[27] E. G. Kalnins, W. Miller, Jr., “Separable components for three-dimensional complex Riemennian spaces”, J. Differential Geom., 14:2 (1979), 221–236 | DOI