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Symmetry reduction of discrete Lagrangian mechanics on Lie groups

Jerrold E. Marsden·Sergey Pekarsky·Steve Shkoller
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Abstract

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $\omega_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.

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Jerrold E. Marsden, Sergey Pekarsky, Steve Shkoller. 2000-04-04. Symmetry reduction of discrete Lagrangian mechanics on Lie groups. https://doi.org/10.1016/s0393-0440(00)00018-8

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