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Regular and Chaotic Dynamics

, Volume 14, Issue 3, pp 389–406 | Cite as

Leonard Euler: Addition theorems and superintegrable systems

  • A. V. TsiganovEmail author
Research Articles

Abstract

We consider the Euler approach to constructing to investigating of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems.

Key words

superintegrable systems addition theorems 

MSC2000 numbers

70H06 70H20 35Q72 

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References

  1. 1.
    Daskaloyannis, C. and Ypsilantis, K., Unified Treatment and Classification of Superintegrable Systems with Integrals Quadratic in Momenta on a Two-Dimensional Manifold, J. Math. Phys., 2006, vol. 47, 042904.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Marquette, I. and Winternitz, P., Superintegrable Systems with Third-Order Integrals of Motion, J. Phys. A, 2008, vol. 30, no. 30, 304031 (10 pp).CrossRefMathSciNetGoogle Scholar
  3. 3.
    Kalnins, E. G., Kress, J. M., and Miller, W. Jr., Nondegenerate 2D Complex Euclidean Superintegrable Systems and Algebraic Varieties, J. Phys. A, 2007, vol. 40, no. 13, pp. 3399–3411.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Euler, L., Institutiones Calculi integralis, Petropoli, 1761 (in Russian: Mosow, 1956).Google Scholar
  5. 5.
    Cayley, A., An Elementary Treatise on Elliptic Functions, London, 1876.Google Scholar
  6. 5a.
    Greenhill, A. G., The Applications of Elliptic Functions, London, 1892.Google Scholar
  7. 6.
    Tsiganov, A.V., On Maximally Superintegrable Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 3, pp. 178–190.CrossRefMathSciNetGoogle Scholar
  8. 7.
    Moser, J., Finitely Many Mass Points on the Line under the Influence of an Exponential Potential — an Integrable System, Dynamical systems, theory and applications (Rencontres, BattelleRes. Inst., Seattle, Wash., 1974), Lect. Notes in Phys., vol. 38, 1975, pp. 467–497.Google Scholar
  9. 8.
    Henrici, A. and Kappeler, T., Global Action-Angle Variables for the Periodic Toda Lattice, Preprint: arXiv:0802.4032.Google Scholar
  10. 9.
    Stäckel, P., Ueber die Integration der Hamilton-Jacobischen Differential Gleichung mittelst Separation der Variabel, Habilitationsschrift, Halle, 1891.Google Scholar
  11. 10.
    Tsiganov, A.V., The Stäckel Systems and Algebraic Curves, J. Math. Phys., 1999, vol. 40, pp. 279–298.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 11.
    Tsiganov, A.V., Duality between Integrable Stäckel Systems, J. Phys. A: Math. Gen., 1999, vol. 32, pp. 7965–7982.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 12.
    Tsiganov, A.V., The Maupertuis Principle and Canonical Transformations of the Extended Phase Space, J. Nonlinear Math. Phys., 2001, vol. 8, no. 1, pp. 157–182.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 13.
    Tsiganov, A.V., Addition Theorem and the Drach Superintegrable Systems, J. Phys. A, 2008, vol. 41, no. 33, 335204 (16 pp).Google Scholar
  15. 14.
    Drach, J., Sur l’intégration logique des équations de la dynamique à deux variables: Forces conservatives. Intêgrales cubiques. Mouvements dans le plan, C.R. Acad. Sci. Paris, 1935, vol. 200, p. 22–26.Google Scholar
  16. 15.
    Tsiganov, A.V., On the Drach Superintegrable Systems, J. Phys. A: Math. Gen., 2000, vol. 33, pp. 7407–7423.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 16.
    Dixon, A. L., On Hyperelliptic Functions of Genus Two, Quart. J. Math., 1904, vol. 36, pp. 1.zbMATHGoogle Scholar
  18. 16a.
    Parker, W.V., Addition Formulas for Hyperelliptic Functions, Bull. Amer. Math. Soc., 1932, vol. 38, pp. 895–901.CrossRefMathSciNetGoogle Scholar
  19. 17.
    Burnside, W. S. and Panton, A.W., Theory of Equations, Harlow: Longmans, 1886.Google Scholar
  20. 18.
    Grigoryev, Yu.A., Khudobakhshov, V.A., and Tsiganov, A.V., On the Euler Superintegrable Systems, to appear.Google Scholar
  21. 19.
    Richelot, F., Über die Integration eines merkwürdigen Systems von Differentialgleichungen, J. Reine Angew. Math., 1842, vol. 23, pp. 354–369.zbMATHGoogle Scholar
  22. 20.
    Kalnins, E. G., Kress, J. M., Pogosyan, G. S., and Miller, W. Jr., Completeness of Superintegrability in Two-Dimensional Constant-Curvature Spaces, J. Phys. A: Math. Gen., 2001, vol. 34, pp. 4705–4720.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 21.
    Darboux, G., Leçons sur la théorie générale des surfaces, Paris, 1898.Google Scholar
  24. 22.
    Koenigs, M.G., Sur les géodésiques a intégrales quadratiques, Note II in G. Darboux, Leçons sur la théorie générale des surfaces, 1898.Google Scholar
  25. 23.
    Grigoryev, Yu.A. and Tsiganov, A.V., Symbolic Software for Separation of Variables in the Hamilton-Jacobi Equation for the L-Systems, Regul. Chaotic Dyn., 2005, vol. 10, no. 4, pp.413–422.CrossRefMathSciNetGoogle Scholar
  26. 24.
    Ibort, A., Magri, F., and Marmo, G., Bihamiltonian Structures and Stäckel Separability, J. Geom. Phys., 2000, vol. 33, pp. 210–228.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 25.
    Benenti, S., Intrinsic Characterization of the Variable Separation in the Hamilton-Jacobi Equation, J. Math. Phys., 1997, vol. 38, pp. 6578–6602.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.St. Petersburg State University PetrodvoretsSt. PetersburgRussia

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