Skip to main content
Log in

On two-dimensional integrable models with a cubic or quartic integral of motion

Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter’s invariance under continuous rescalings and a dihedral symmetry, we construct new integrable models with a cubic or quartic integral, each of which involves either one or two continuous parameters. A reducible case related to the two-dimensional wave equation is discussed as well. We conjecture a hidden D 2n dihedral symmetry for models with an integral of nth order in the velocities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. S.V. Kovalevskaya, Sur le probleme de la rotation dun corps solide autour dun point fixe, Acta Math. 12 (1889) 177.

    Article  MathSciNet  Google Scholar 

  2. D.N. Goryachev, A new case of the integrability of Eulers dynamical equations, Varshav. Univ. Izv. 11 (1916) 3.

    Google Scholar 

  3. M. Hénon, Integrals of the Toda lattice, Phys. Rev. B 9 (1974) 1921.

    Article  ADS  Google Scholar 

  4. A.S. Fokas and P.A. Lagerstrom, Quadratic and cubic invariants in classical mechanics, J. Math. Anal. Appl. 74 (1980) 325.

    Article  MathSciNet  MATH  Google Scholar 

  5. C.R. Holt, Construction of new integrable Hamiltonians in two degrees of freedom, J. Math. Phys. 23 (1982) 1037.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. V.I. Inozemtsev, New integrable classical system with two degrees of freedom, Phys. Lett. A 96 (1983) 447.

    Article  MathSciNet  ADS  Google Scholar 

  7. L.S. Hall, A theory of exact and approximate configurational invariants, Physica D 8 (1983) 90.

    ADS  Google Scholar 

  8. B. Grammaticos, B. Dorizzi and A. Ramani, Integrability of Hamiltonians with third- and fourth-degree polynomial potentials, J. Math. Phys. 24 (1983) 2289.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. G. Thompson, Polynomial constants of motion in flat space, J. Math. Phys. 25 (1984) 3474.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. M. Wojciechowska and S. Wojciechowski, New integrable potentials of two degrees of freedom, Phys. Lett. A 105 (1984) 11.

    Article  MathSciNet  ADS  Google Scholar 

  11. R.S. Kaushal, S.C. Mishra and K.C. Tripathy, Construction of the second constant of motion for two-dimensional classical systems, J. Math. Phys. 26 (1985) 420.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. P.G.L. Leach, Comment on an aspect of a paper by G. Thompson, J. Math. Phys. 27 (1986) 153.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. H.M. Yehia, On the integrability of certain problems in particle and rigid body dynamics, J. Mecan. Theor. Appl. 5 (1986) 55.

    MathSciNet  ADS  MATH  Google Scholar 

  14. J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rept. 147 (1987) 87 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. T. Sen, Integrable potentials with cubic and quartic invariants, Phys. Lett. A 122 (1987) 100.

    Article  ADS  Google Scholar 

  16. T. Sen, A class of integrable potentials, J. Math. Phys. 28 (1987) 2841 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. N.W. Evans, On Hamiltonian systems in two degrees of freedom with invariants quartic in the momenta of form \( p_1^2p_2^2 \) · · · , J. Math. Phys. 31 (1990) 600.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. G. Bozis, Two-dimensional integrable potentials with quartic invariants, J. Phys. A 25 (1992) 3329.

    MathSciNet  ADS  Google Scholar 

  19. E.N. Selivanova, New families of conservative systems on S 2 possessing an integral of fourth degree in momenta, Ann. Glob. Anal. Geom. 17 (1999) 201 [dg-ga/9712018].

    Article  MathSciNet  MATH  Google Scholar 

  20. E.N. Selivanova, On the case of Goryachev-Chaplygin and new examples of integrable conservative systems on S 2, Commun. Math. Phys. 207 (1999) 641 [math/9801124].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. K.P. Hadeler and E.N. Selivanova, On the case of Kovalevskaya and new examples of integrable conservative systems on S 2, Reg. Chaot. Dyn. 4 (1999) 45 [math/9801063].

    Article  MathSciNet  MATH  Google Scholar 

  22. H.M. Yehia, New generalizations of all the known integrable problems in rigid-body dynamics, J. Phys. A 32 (1999) 7565.

    MathSciNet  ADS  Google Scholar 

  23. M. Karlovini, G. Pucacco, K. Rosquist and L. Samuelsson, A unified treatment of quartic invariants at fixed and arbitrary energy, J. Math. Phys. 43 (2002) 4041.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. H.M. Yehia, On certain two-dimensional conservative mechanical systems with a cubic second integral, J. Phys. A 35 (2002) 9469.

    MathSciNet  ADS  Google Scholar 

  25. H.M. Yehia, Kovalevskayas integrable case: generalizations and related new results, Reg. Chaot. Dyn. 8 (2003) 337.

    Article  MathSciNet  MATH  Google Scholar 

  26. K. Nakagawa, A. Maciejewski and M. Przybylska, New integrable Hamiltonian system with first integral quartic in momenta, Phys. Lett. A 343 (2005) 171.

    Article  MathSciNet  ADS  Google Scholar 

  27. H.M. Yehia, The master integrable two-dimensional system with a quartic second integral, J. Phys. A 39 (2006) 5807.

    MathSciNet  ADS  Google Scholar 

  28. G. Valent, Explicit integrable systems on two dimensional manifolds with a cubic first integral, Commun. Math. Phys. 299 (2010) 631 [arXiv:1002.2033].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. H.M. Yehia and A.A. Elmandouh, New conditional integrable cases of motion of a rigid body with Kovalevskayas configuration, J. Phys. A 44 (2011) 012001.

    MathSciNet  ADS  Google Scholar 

  30. H.M. Yehia, A new 2D integrable system with a quartic second invariant, J. Phys. A 45 (2012) 5209 [arXiv:1208.4829].

    MathSciNet  Google Scholar 

  31. A. Galajinsky, Remark on integrable deformations of the Euler top, arXiv:1211.0760.

  32. H.M. Yehia and A.A. Elmandouh, A new integrable problem with a quartic integral in the dynamics of a rigid body, J. Phys. A 46 (2013) 142001 [arXiv:1302.7057].

    MathSciNet  ADS  Google Scholar 

  33. G. Valent, On a class of integrable systems with a quartic first integral, Reg. Chaot. Dyn. 18 (2013) 394 [arXiv:1304.5859].

    Article  MathSciNet  MATH  Google Scholar 

  34. H.M. Yehia and A.M. Hussein, New families of integrable two-dimensional systems with quartic second integrals, arXiv:1308.1442.

  35. Y. Aizawa and N. Saito, On the stability of isolating integrals. I. Effect of the perturbation in the potential function, J. Phys. Soc. Jap. 32 (1972) 1636.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Olaf Lechtenfeld.

Additional information

ArXiv ePrint: 1306.5238

Rights and permissions

Reprints and permissions

About this article

Cite this article

Galajinsky, A., Lechtenfeld, O. On two-dimensional integrable models with a cubic or quartic integral of motion. J. High Energ. Phys. 2013, 113 (2013). https://doi.org/10.1007/JHEP09(2013)113

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP09(2013)113

Keywords

Navigation