The motion of a rigid body in a quadratic potential: an integrable discretization

Details The motion of a rigid body in a quadratic potential: an integrable discretization The motion of a rigid body in a quadratic potential: an integrable discretization

1999-09-14

arxiv.org/abs/solv-int/9909009v1

Intern. Math. Research Notices, 2000, No 12, p.643-663.

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTeX, 15 pp

Scientific paper

The motion of a rigid body in a quadratic potential is an important example of an integrable Hamiltonian system on a dual to a semidirect product Lie algebra so(n) x Symm(n). We give a Lagrangian derivation of the corresponding equations of motion, and introduce a discrete time analog of this system. The construction is based on the discrete time Lagrangian mechanics on Lie groups, accompanied with the discrete time Lagrangian reduction. The resulting multi-valued map (correspondence) on the dual to so(n) x Symm(n) is Poisson with respect to the Lie-Poisson bracket, and is also completely integrable. We find a Lax representation based on matrix factorisations, in the spirit of Veselov-Moser.

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