Symmetry reduction of discrete Lagrangian mechanics on Lie groups

TL;DR

This work develops a principled framework for symmetry reduction of discrete Lagrangian mechanics on Lie groups. By constructing a $G$-invariant discrete Lagrangian $\\mathbb{L}$ on $G \times G$ and reducing with respect to $G$, the authors obtain DEP and DLP algorithms that preserve a Poisson structure on a neighborhood of $G$, realized as the pull-back of the Lie-Poisson form via the reduced Legendre transform $F \\ell$. The Poisson structure is shown to be compatible with Weinstein’s groupoid/algebroid picture, and the DEP flow preserves its symplectic leaves, yielding dynamically invariant manifolds for the reduced dynamics. The DEP/ DLP correspondence is interpreted as a discrete Lie–Poisson Hamilton–Jacobi theory, connecting discrete reduction to continuous reduced dynamics. The rigid body example on $SO(3)$ illustrates the construction: the reduced Lagrangian $\\ell(f)$, the Legendre transform $F \\ell$, and the resulting Casimir functions define invariant leaves that organize the reduced flow in a geometrically transparent way.

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 $ω_\mathbb{L}$ 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.

Theorems & Definitions (10)

• Remark 2.1
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Theorem 3.1
• proof
• Corollary 3.1
• proof
• Theorem 3.2
• proof