Doubly Exotic $N$th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

İsmet Yurduşen ; Adrián Mauricio Escobar-Ruiz ; Irlanda Palma y Meza Montoya

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ are explored. The study is restricted to Hamiltonians allowing separation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. In particular, the Hamiltonian $\mathcal H$ admits a polynomial integral of order $N>2$. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case $N=5$, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case $N>2$ and a formulation of inverse problem in superintegrability are briefly discussed as well.

Détail

Keywords: integrability in classical mechanics, higher-order superintegrability, separation of variables, exotic potentials.

Bibliographie

[1] Abouamal I., Winternitz P., “Fifth-order superintegrable quantum systems separating in Cartesian coordinates: doubly exotic potentials”, J. Math. Phys., 59 (2018), 022104, 27 pp., arXiv: 1708.03379 | DOI | MR | Zbl

[2] Baldwin D., Hereman W., PainleveTestV4-2018.m: A Mathematica package for the Painlevé test of systems of nonlinear ordinary and partial differential equations, 2018, https://inside.mines.edu/w̃hereman/software/painleve/mathematica/ | MR

[3] Bertrand J.L.F., “Théorème relatif au mouvement d'un point attiré vers un centre fixe”, C. R. Acad. Sci. Paris, 77 (1873), 849–853 | Zbl

[4] Bertrand S., Kubu O., Šnobl L., “On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields”, J. Phys. A: Math. Theor., 54 (2021), 015201, 27 pp., arXiv: 2008.01987 | DOI | MR

[5] Bérubé J., Winternitz P., “Integrable and superintegrable quantum systems in a magnetic field”, J. Math. Phys., 45 (2004), 1959–1973, arXiv: math-ph/0311051 | DOI | MR | Zbl

[6] Chanu C.M., Degiovanni L., Rastelli G., “The Tremblay–Turbiner–Winternitz system as extended Hamiltonian”, J. Math. Phys., 55 (2014), 122701, 8 pp., arXiv: 1404.4825 | DOI | MR | Zbl

[7] Chanu C.M., Rastelli G., “Extended Hamiltonians and shift, ladder functions and operators”, Ann. Physics, 386 (2017), 254–274, arXiv: 1705.09519 | DOI | MR | Zbl

[8] Daskaloyannis C., “Generalized deformed oscillator and nonlinear algebras”, J. Phys. A: Math. Gen., 24 (1991), L789–L794 | DOI | MR | Zbl

[9] Désilets J.F., Winternitz P., Yurduşen İ., “Superintegrable systems with spin and second-order integrals of motion”, J. Phys. A: Math. Theor., 45 (2012), 475201, 26 pp., arXiv: 1208.2886 | DOI | MR | Zbl

[10] Dorizzi B., Grammaticos B., Ramani A., Winternitz P., “Integrable Hamiltonian systems with velocity-dependent potentials”, J. Math. Phys., 26 (1985), 3070–3079 | DOI | MR | Zbl

[11] Drach J., “Sur l'integration logique des équations de la dynamique à deux variables: Forces constructives. Intégrales cubiques. Mouvements dans le plan”, C. R. Acad. Sci. Paris, 200 (1935), 22–26 | MR

[12] Drach J., “Sur l'integration logique et sur la tranformation des équations de la dynamique à deux variables: Forces conservatives. Intégrales cubiques. Mouvements dans le plan”, C. R. Acad. Sci. Paris, 200 (1935), 599–602

[13] Escobar-Ruiz A.M., Linares R., Winternitz P., “New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates”, J. Phys. A: Math. Theor., 53 (2020), 445203, 26 pp., arXiv: 2004.12173 | DOI | MR

[14] Escobar-Ruiz A.M., López Vieyra J.C., Winternitz P., “Fourth order superintegrable systems separating in polar coordinates. I Exotic potentials”, J. Phys. A: Math. Theor., 50 (2017), 495206, 35 pp., arXiv: 1706.08655 | DOI | MR | Zbl

[15] Escobar-Ruiz A.M., López Vieyra J.C., Winternitz P., Yurduşen İ., “Fourth-order superintegrable systems separating in polar coordinates. II Standard potentials”, J. Phys. A: Math. Theor., 51 (2018), 455202, 24 pp., arXiv: 1804.05751 | DOI | MR | Zbl

[16] Escobar-Ruiz A.M., Winternitz P., Yurduşen İ., “General $N$th-order superintegrable systems separating in polar coordinates”, J. Phys. A: Math. Theor., 51 (2018), 40LT01, 12 pp., arXiv: 1806.06849 | DOI | MR | Zbl

[17] Evans N.W., “Super-integrability of the Winternitz system”, Phys. Lett. A, 147 (1990), 483–486 | DOI | MR

[18] Evans N.W., “Superintegrability in classical mechanics”, Phys. Rev. A, 41 (1990), 5666–5676 | DOI | MR

[19] Fradkin D.M., “Three-dimensional isotropic harmonic oscillator and ${\rm SU}_{3}$”, Amer. J. Phys., 33 (1965), 207–211 | DOI | MR | Zbl

[20] Friš J., Mandrosov V., Smorodinsky Y.A., Uhlíř M., Winternitz P., “On higher symmetries in quantum mechanics”, Phys. Lett., 16 (1965), 354–356 | DOI | MR

[21] Goldstein H., Classical mechanics, Addison-Wesley Series in Physics, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1980 | MR | Zbl

[22] Gravel S., “Hamiltonians separable in Cartesian coordinates and third-order integrals of motion”, J. Math. Phys., 45 (2004), 1003–1019, arXiv: math-ph/0302028 | DOI | MR | Zbl

[23] Gravel S., Winternitz P., “Superintegrability with third-order integrals in quantum and classical mechanics”, J. Math. Phys., 43 (2002), 5902–5912, arXiv: math-ph/0206046 | DOI | MR | Zbl

[24] Grigoriev Y.A., Tsiganov A.V., “On superintegrable systems separable in Cartesian coordinates”, Phys. Lett. A, 382 (2018), 2092–2096, arXiv: 1712.07321 | DOI | MR | Zbl

[25] Grosche C., Pogosyan G.S., Sissakian A.N., “Path integral discussion for Smorodinsky–Winternitz potentials. II The two- and three-dimensional sphere”, Fortschr. Phys., 43 (1995), 523–563, arXiv: hep-th/9402121 | DOI | MR | Zbl

[26] Grosche C., Pogosyan G.S., Sissakian A.N., “Path integral approach to superintegrable potentials. Two-dimensional hyperboloid”, Phys. Part. Nuclei, 27 (1996), 244–278

[27] Grosche C., Pogosyan G.S., Sissakian A.N., “Path integral discussion for superintegrable potentials. IV. Three-dimensional pseudosphere”, Phys. Part. Nuclei, 28 (1997), 486–519 | DOI

[28] Güngör F., Kuru S., Negro J., Nieto L.M., “Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates”, Nonlinearity, 30 (2017), 1788–1808, arXiv: 1411.6216 | DOI | MR | Zbl

[29] Hietarinta J., “Direct methods for the search of the second invariant”, Phys. Rep., 147 (1987), 87–154 | DOI | MR

[30] Jauch J.M., Hill E.L., “On the problem of degeneracy in quantum mechanics”, Phys. Rev., 57 (1940), 641–645 | DOI | MR | Zbl

[31] 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

[32] Kalnins E.G., Kress J.M., Miller Jr. W., “Second-order superintegrable systems in conformally flat spaces. V Two- and three-dimensional quantum systems”, J. Math. Phys., 47 (2006), 093501, 25 pp. | DOI | MR | Zbl

[33] Kalnins E.G., Kress J.M., Miller Jr. W., “Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties”, J. Phys. A: Math. Theor., 40 (2007), 3399–3411, arXiv: 0708.3044 | DOI | MR | Zbl

[34] Kalnins E.G., Kress J.M., Miller Jr. W., Separation of variables and superintegrability. The symmetry of solvable systems, IOP Expanding Physics, IOP Publishing, Bristol, 2018 | DOI | MR | Zbl

[35] Kalnins E.G., Kress J.M., Miller Jr. W., Post S., “Structure theory for second order 2D superintegrable systems with 1-parameter potentials”, SIGMA, 5 (2009), 008, 24 pp., arXiv: 0901.3081 | DOI | MR | Zbl

[36] Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., “Superintegrable systems in Darboux spaces”, J. Math. Phys., 44 (2003), 5811–5848, arXiv: math-ph/0307039 | DOI | MR | Zbl

[37] Kalnins E.G., Kress J.M., Winternitz P., “Superintegrability in a two-dimensional space of nonconstant curvature”, J. Math. Phys., 43 (2002), 970–983, arXiv: math-ph/0108015 | DOI | MR | Zbl

[38] Kalnins E.G., Miller Jr. W., Pogosyan G.S., “Nondegenerate superintegrable systems in $n$-dimensional spaces of constant curvature”, Phys. Atomic Nuclei, 70 (2007), 545–553, arXiv: 0912.2278 | DOI

[39] Kalnins E.G., Williams G.C., Miller Jr. W., Pogosyan G.S., “On superintegrable symmetry-breaking potentials in $N$-dimensional Euclidean space”, J. Phys. A: Math. Gen., 35 (2002), 4755–4773 | DOI | MR | Zbl

[40] Laplace P.S., Traité de mécanique céleste, Duprat, Paris, 1799

[41] Lenz W., “Über den Bewegungsverlauf und die Quantenzustände der gestörten Keplerbewegung”, Z. Phys., 24 (1924), 197–207 | DOI

[42] Makarov A.A., Smorodinsky J.A., Valiev K., Winternitz P., “A systematic search for nonrelativistic systems with dynamical symmetries”, Nuovo Cimento A, 52 (1967), 1061–1084 | DOI

[43] Marchesiello A., Šnobl L., Winternitz P., “Three-dimensional superintegrable systems in a static electromagnetic field”, J. Phys. A: Math. Theor., 48 (2015), 395206, 24 pp., arXiv: 1507.04632 | DOI | MR | Zbl

[44] Marchesiello A., Šnobl L., Winternitz P., “Spherical type integrable classical systems in a magnetic field”, J. Phys. A: Math. Theor., 51 (2018), 135205, 24 pp. | DOI | MR | Zbl

[45] Marquette I., “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II Painlevé transcendent potentials”, J. Math. Phys., 50 (2009), 095202, 18 pp., arXiv: 0811.1568 | DOI | MR | Zbl

[46] Marquette I., “Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems”, J. Math. Phys., 53 (2012), 012901, 12 pp., arXiv: 1109.4471 | DOI | MR | Zbl

[47] Marquette I., Sajedi M., Winternitz P., “Fourth order superintegrable systems separating in Cartesian coordinates I Exotic quantum potentials”, J. Phys. A: Math. Theor., 50 (2017), 315201, 29 pp., arXiv: 1703.09751 | DOI | MR | Zbl

[48] Marquette I., Winternitz P., “Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion”, J. Math. Phys., 48 (2007), 012902, 16 pp., arXiv: math-ph/0608021 | DOI | MR | Zbl

[49] Miller Jr. W., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, 4, Addison-Wesley Publishing Co., Reading, Mass. – London – Amsterdam, 1977 | MR | Zbl

[50] Miller Jr. W., “Second order superintegrable systems in three dimensions”, SIGMA, 1 (2005), 015, 17 pp., arXiv: nlin.SI/0511035 | DOI | MR | Zbl

[51] Miller Jr. W., Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[52] Nikitin A.G., “Higher-order symmetry operators for Schrödinger equation”, Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, 37, Amer. Math. Soc., Providence, RI, 2004, 137–144, arXiv: 1603.01715 | DOI | MR | Zbl

[53] Nikitin A.G., “Matrix superpotentials and superintegrable systems for arbitrary spin”, J. Phys. A: Math. Theor., 45 (2012), 225205, 13 pp., arXiv: 1201.4929 | DOI | MR | Zbl

[54] Nikitin A.G., “New exactly solvable systems with Fock symmetry”, J. Phys. A: Math. Theor., 45 (2012), 485204, 9 pp., arXiv: 1205.3094 | DOI | MR | Zbl

[55] Popper I., Post S., Winternitz P., “Third-order superintegrable systems separable in parabolic coordinates”, J. Math. Phys., 53 (2012), 062105, 20 pp., arXiv: 1204.0700 | DOI | MR | Zbl

[56] Post S., Winternitz P., “An infinite family of superintegrable deformations of the Coulomb potential”, J. Phys. A: Math. Theor., 43 (2010), 222001, 11 pp., arXiv: 1003.5230 | DOI | MR | Zbl

[57] Post S., Winternitz P., “General $N$th order integrals of motion in the Euclidean plane”, J. Phys. A: Math. Theor., 48 (2015), 405201, 24 pp., arXiv: 1501.00471 | DOI | MR | Zbl

[58] Rañada M.F., “Superintegrable $n=2$ systems, quadratic constants of motion, and potentials of Drach”, J. Math. Phys., 38 (1997), 4165–4178 | DOI | MR | Zbl

[59] Rodriguez M.A., Winternitz P., “Quantum superintegrability and exact solvability in $n$ dimensions”, J. Math. Phys., 43 (2002), 1309–1322, arXiv: math-ph/0110018 | DOI | MR | Zbl

[60] Runge C., Vektoranalysis, v. I, Hirzel, Leipzig, 1919 | Zbl

[61] Thompson G., “Polynomial constants of motion in flat space”, J. Math. Phys., 25 (1984), 3474–3478 | DOI | MR | Zbl

[62] Tremblay F., Turbiner A.V., Winternitz P., “An infinite family of solvable and integrable quantum systems on a plane”, J. Phys. A: Math. Theor., 42 (2009), 242001, 10 pp., arXiv: 0904.0738 | DOI | MR | Zbl

[63] Tremblay F., Turbiner A.V., Winternitz P., “Periodic orbits for an infinite family of classical superintegrable systems”, J. Phys. A: Math. Theor., 43 (2010), 015202, 14 pp., arXiv: 0910.0299 | DOI | MR | Zbl

[64] Tremblay F., Winternitz P., “Third-order superintegrable systems separating in polar coordinates”, J. Phys. A: Math. Theor., 43 (2010), 175206, 17 pp., arXiv: 1002.1989 | DOI | MR | Zbl

[65] Tsiganov A.V., “The Drach superintegrable systems”, J. Phys. A: Math. Gen., 33 (2000), 7407–7422, arXiv: nlin.SI/0001053 | DOI | MR | Zbl

[66] Tsiganov A.V., “Addition theorems and the Drach superintegrable systems”, J. Phys. A: Math. Theor., 41 (2008), 335204, 16 pp., arXiv: 0805.3443 | DOI | MR | Zbl

[67] Tsiganov A.V., “On maximally superintegrable systems”, Regul. Chaotic Dyn., 13 (2008), 178–190, arXiv: 0711.2225 | DOI | MR | Zbl

[68] Winternitz P., Yurduşen İ., “Integrable and superintegrable systems with spin”, J. Math. Phys., 47 (2006), 103509, 10 pp., arXiv: math-ph/0604050 | DOI | MR | Zbl

[69] Winternitz P., Yurduşen İ., “Integrable and superintegrable systems with spin in three-dimensional Euclidean space”, J. Phys. A: Math. Theor., 42 (2009), 385203, 20 pp., arXiv: 0906.1021 | DOI | MR | Zbl

[70] Yurduşen İ., Tuncer O.O., Winternitz P., “Superintegrable systems with spin and second-order tensor and pseudo-tensor integrals of motion”, J. Phys. A: Math. Theor., 54 (2021), 305201, 32 pp., arXiv: 2104.07436 | DOI | MR