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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 345
Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top A.I.Bobenko and Yu.B.Suris
We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrablesystem of classical mechanics, -- the Lagrange top. We recall the derivationof the Euler--Poinsot equations of motion both in the frame moving withthe body and in the rest frame (the latter ones being less widely known).We find a discrete time Lagrange function turning into the known continuoustime Lagrangian in the continuous limit, and elaborate both descriptions ofthe resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuoustime system, the integrals of motion being deformed. The discrete time Lax representations are also found. Kirchhoff's kinetic analogy between elastic curves and motions of the Lagrange top is also generalised to the discrete context.
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