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International Mathematics Research Notices (2000) 2000:643-663, doi:10.1155/S1073792800000350
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

The Motion of a Rigid Body in a Quadratic Potential: An Integrable Discretization

Yuri B. Suris

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 Formula. 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 Formula 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 and Moser.


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