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

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Abstract

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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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