Symmetry reduction of discrete Lagrangian mechanics on Lie groups

Journal article

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.

Authors: Marsden, J; Pekarsky, S; Shkoller, S

• Journal: Journal of Geometry and Physics
• Volume: 36
• Issue: 1-2
• Pages: 140-151
• Publication date: 2000-04-04
• DOI: 10.1016/S0393-0440(00)00018-8
• ISSN: 0393-0440
• Keywords: math.SG; math.NA