Symmetry reduction of discrete Lagrangian mechanics on Lie groups
Jerrold E. Marsden, Sergey Pekarsky, Steve Shkoller
对于由（约化）拉格朗日量ℓ确定的李群G上的离散力学系统，我们通过相应勒让德变换将李代数对偶𝔤^*上的李-泊松结构拉回，定义了一个泊松结构。本文证明的主要结果是，该结构与在G × G上规范离散拉格朗日2-形式ω_𝕃在对称群G下的约化相一致。其辛叶随后成为约化离散系统的动态不变流形。我们还建立了我们的方法与群胚和代数胚方法以及约化哈密顿-雅可比方程之间的联系。文中以刚体为例进行了讨论。
For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ 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 dis...
