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Degenerate integrable systems on the plane with a cubic integral of motion

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Abstract

We propose a classification of the known two-dimensional Hamiltonian systems of the natural form possessing an additional integral of motion that is cubic in the momenta. For degenerate systems of the Stäckel type, the additional cubic integral has the form of a “generalized angular momentum”. This allows constructing n-dimensional degenerate systems of the Stäckel type with additional cubic integrals of motion.

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References

  1. Ya. A. Smorodinsky, M. Uhlir, P. Winternitz, and I. Fris,Sov. J. Nucl. Phys.,4, 444 (1967).

    Google Scholar 

  2. N. W. Evans,Phys. Rev. A,41, 5666 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  3. D. Bonatos, C. Daskaloyannis, and K. Kokkotas,Phys. Rev. A.,50, 3700 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  4. P. Letourneau and L. Vinet,Ann. Phys.,243, 144 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  5. V. I. Arnold,Mathematical Methods of Classical Mechanics [in Russian], (3rd ed.) Nauka, Moscow (1989); English transl. prev. ed., Springer, Berlin (1978).

    Google Scholar 

  6. A. M. Perelomov,Integrable Systems of Classical Mechanics and Lie Algebras [in Russian], Nauka, Moscow (1990); English transl. Birkhäuser, Basel (1990).

    Google Scholar 

  7. E. T. Whittaker,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, (With an Introduction to the Problem of 3 Bodies), Cambridge Univ. Press, Cambridge (1927).

    MATH  Google Scholar 

  8. P. Stäckel,Compt. Rend. Acad. Sci. (Paris),116, 485, 1284 (1893).

    Google Scholar 

  9. P.-S. Laplace,Traité de Mechanique Céleste, Vol. 1, Paris (1799).

  10. J. Drach,Compt. Rend. Acad. Sci., (Paris),200, 22 (1935).

    Google Scholar 

  11. A. S. Fokas and P. Lagerstrom,J. Math. Anal. Appl.,74, 325 (1980).

    Article  MathSciNet  Google Scholar 

  12. C. R. Holt,J. Math. Phys.,23, 37 (1982).

    Article  MathSciNet  Google Scholar 

  13. G. Thompson,J. Math. Phys.,25, 3474 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Hietarinta,Phys. Rep.,147, 87 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  15. M. Karlovini and K. Rosquist, “A unified treatment of cubic invariants at fixed and arbitrary energy,” Preprint solv-int/9811005 (1998).

  16. A. V. Tsyganov,Theor. Math. Phys.,115, 377 (1998).

    MathSciNet  Google Scholar 

  17. A. V. Tsyganov,Theor. Math. Phys.,120, 840 (1999).

    MathSciNet  Google Scholar 

  18. A. V. Tsiganov, “Canonical transformations of the extended phase space, Toda lattices and Stäckel family of integrable systems,” Preprint solv-int/9909006 (1999).

  19. A. V. Tsiganov “Canonical transformations of the time for the Toda lattice and the Holt system,” Preprint solv-int/99011001 (1999).

  20. A. V. Tsiganov,J. Phys. A,32, 7983 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  21. P. Vanhaecke,Math. Z.,211, 265 (1992).

    MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    A. V. Tsyganov

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  1. A. V. Tsyganov
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 426–444, September, 2000.

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Tsyganov, A.V. Degenerate integrable systems on the plane with a cubic integral of motion. Theor Math Phys 124, 1217–1233 (2000). https://doi.org/10.1007/BF02551000

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  • DOI: https://doi.org/10.1007/BF02551000

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