Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 031, 9 pages      arXiv:1204.1801
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Superintegrable Stäckel Systems on the Plane: Elliptic and Parabolic Coordinates

Andrey V. Tsiganov
St. Petersburg State University, St. Petersburg, Russia

Received April 10, 2012, in final form May 21, 2012; Published online May 25, 2012

Recently we proposed a generic construction of the additional integrals of motion for the Stäckel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for elliptic and parabolic coordinate systems on the plane.

Key words: integrability; superintegrability; separation of variables; Abel equations; addition theorems.

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  1. Baker H.F., Abel's theorem and the allied theory including the theory of the theta functions, Cambridge University Press, Cambridge, 1897.
  2. Borisov A.V., Kilin A.A., Mamaev I.S., Superintegrable system on a sphere with the integral of higher degree, Regul. Chaotic Dyn. 14 (2009), 615-620.
  3. Euler L., Institutiones Calculi integralis, Acta Petropolitana, 1761.
  4. Grigoryev Y.A., Khudobakhshov V.A., Tsiganov A.V., On Euler superintegrable systems, J. Phys. A: Math. Theor. 42 (2009), 075202, 11 pages.
  5. Kalnins E.G., Separation of variables for Riemannian spaces of constant curvature, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 28, Longman Scientific & Technical, Harlow, 1986.
  6. Kalnins E.G., Miller W., Structure theory for extended Kepler-Coulomb 3D classical superintegrable systems, arXiv:1202.0197.
  7. Maciejewski A.J., Przybylska M., Tsiganov A.V., On algebraic construction of certain integrable and super-integrable systems, Phys. D 240 (2011), 1426-1448, arXiv:1011.3249.
  8. Popperi I., Post S., Winternitz P., Third-order superintegrable systems separable in parabolic coordinates, arXiv:1204.0700.
  9. Post S., Winternitz P., A nonseparable quantum superintegrable system in 2D real Euclidean space, J. Phys. A: Math. Theor. 44 (2011), 162001, 8 pages, arXiv:1101.5405.
  10. Richelot F., Ueber die Integration eines Merkwürdigen Systems von Differentialgleichungen, J. Reine Angew. Math. 23 (1842), 354-369.
  11. Stäckel P., Über die Integration der Hamilton-Jacobischen Differential Gleichung Mittelst Separation der Variabeln, Habilitationsschrift, Halle, 1891.
  12. Tsiganov A.V., Addition theorems and the Drach superintegrable systems, J. Phys. A: Math. Theor. 41 (2008), 335204, 16 pages, arXiv:0805.3443.
  13. Tsiganov A.V., Leonard Euler: addition theorems and superintegrable systems, Regul. Chaotic Dyn. 14 (2009), 389-406, arXiv:0810.1100.
  14. Tsiganov A.V., On maximally superintegrable systems, Regul. Chaotic Dyn. 13 (2008), 178-190, arXiv:0711.2225.
  15. Tsiganov A.V., On the superintegrable Richelot systems, J. Phys. A: Math. Theor. 43 (2010), 055201, 14 pages, arXiv:0909.2923.

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