Symmetry reduction of discrete Lagrangian mechanics on Lie groups

✍ Scribed by Jerrold E. Marsden; Sergey Pekarsky; Steve Shkoller

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 91 KB
• Volume: 36
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(00)00018-8

✦ Synopsis

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian , we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra 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 ω L on G × 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.