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On the euler equation: Bi-Hamiltonian structure and integrals in involution

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We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the ‘physical’ phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.

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  1. 1.

    Arnol'd, V. I.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), 319–361.

  2. 2.

    Adler, M., vanMoerbeke, P.: Completely integrable systems, Kac-Moody Lie algebras and curves, Adv. in Math. 38 (1980), 267–317.

  3. 3.

    Dikii, L. A.: Hamiltonian systems connected with the rotation group, Funct. Anal. Appl. 6 (1972), 326–327.

  4. 4.

    Libermann, P. and Marle, C. M.: Symplectic Geometry and Analytical Mechanics, D. Reidel, Dordrecht, 1987.

  5. 5.

    Magri, F.: A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162.

  6. 6.

    Manakov, S. V.: Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976), 328–329.

  7. 7.

    Meshcheryakov, M. V.: A characteristic property of the inertia tensor of a multi-dimensional rigid body, Russian Math. Surveys 38 (1983), 156–157.

  8. 8.

    Mishchenko, A. S., Fomenko, A. T.: Euler's equations on finite-dimensional Lie groups, Math. USSR Izv. 12 (1978), 371–389.

  9. 9.

    Mishchenko, A. S.: Integral geodesics of a flow on Lie groups, Funct. Anal. Appl. 4 (1970), 232–235.

  10. 10.

    Ratiu, T.: The motion of the free n-dimensional rigid body, Indiana Univ. Math. J. 29 (1980), 609–629.

  11. 11.

    Reyman, A. G. and Semenov-Tian-Shansky, M. A.: Group theoretical methods in the theory of finite dimensional integrable systems, in V. I.Arnol'd and S. P.Novikov (eds), Dynamical Systems VII, Springer, Berlin, 1994, pp. 116–225.

  12. 12.

    Ugaglia, M.: Sistemi dinamici integrabili su algebre di Lie: funzioni di Casimir e significato del parametro spettrale, Thesis, Phys. Dept., Univ. of Torino, 1994.

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Morosi, C., Pizzocchero, L. On the euler equation: Bi-Hamiltonian structure and integrals in involution. Letters in Mathematical Physics 37, 117–135 (1996). https://doi.org/10.1007/BF00416015

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Mathematics Subject Classifications (1991)

  • 58F05
  • 58F07
  • 70Exx
  • 70Hxx

Key words

  • integrable systems
  • Euler equations on Lie algebras
  • Hamiltonian structures