General Relativity and Gravitation

, Volume 38, Issue 6, pp 1115–1127 | Cite as

Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket

Research Article

Abstract

Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket e(3) includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Among the applications with a Lie-Poisson bracket so(4) and so(3, 1) is the description of free rigid body motion in a space of constant curvature. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the classic search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.

Keywords

Hamiltonian systems Integrability Computer algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, M., van Moerbeke, P.: Adv. Math. Suppl. Stud. 9, 81 (1986)MathSciNetGoogle Scholar
  2. 2.
    Reyman, A.G., Semenov-Tian-Shansky, M.A.: Comm. Math. Phys. 105, 461 (1986)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Sklyanin, E.K.: J. Phys. A. 21, 2375 (1988)CrossRefMATHMathSciNetADSGoogle Scholar
  4. 4.
    Sokolov, V.V.: Theor. Math. Physics 129(1), 31 (2001)MATHGoogle Scholar
  5. 5.
    Borisov, A.V., Mamaev, S.I., Sokolov V.V.: Doklady RAN 381(5), 614 (2001)MathSciNetGoogle Scholar
  6. 6.
    Sokolov, V.V.: Doklady RAN, 394(5) 602 (2004)Google Scholar
  7. 7.
    Sokolov, V.V., Wolf, T.: Integrable quadratic Hamiltonians on so(4) and so(3, 1) J. Phys. A: Math. Gen. 39, 1915–1926 (2006) [and arXiv nlin. SI/0405066]Google Scholar
  8. 8.
    Tsiganov, A.V.: Reg. Chaot. Dyn. 7(3), 331 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Tsiganov, A.V., Goremykin, O.V.: Integrable systems on so(4) related with XXX spin chains with boundaries. J. Phys. A 37, 4843–4849 (2004)Google Scholar
  10. 10.
    Sakovich, S.Yu.: (2004) [nlin.SI/0408027]Google Scholar
  11. 11.
    Wolf, T., Efimovskaya, O.V.: Regular and Chaotic Dynamics 8, 155 (2003)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Wolf, T.: Applications of crack in the classification of integrable systems. CRM Proceedings and Lecture Notes 37, 283–300 (2004) [arXiv nlin.SI/0301032]Google Scholar
  13. 13.
    Wolf, T.: J. Symb. Comp. 33, 367 (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science and Business Media Inc., New York 2006

Authors and Affiliations

  1. 1.Department of MathematicsBrock UniversityOntarioCanada

Personalised recommendations