Discrete Euler-Poincaré and Lie-Poisson Equations
Jerrold E. Marsden, Sergey Pekarsky, Steve Shkoller
TL;DR
The paper develops a discrete, structure-preserving framework for Euler–Poincaré and Lie–Poisson reductions on Lie groups by constructing a $G$-invariant discrete Lagrangian ${\mathbb L}: G \times G \to \mathbb{R}$ and reducing it to a reduced Lagrangian $\ell: G \to \mathbb{R}$. This yields the discrete Euler-Poincaré (DEP) equations on $G$ and a discrete Lie-Poisson (DLP) flow on ${\mathfrak g}^*$, with reconstruction linking back to the discrete Euler–Lagrange equations on $G \times G$ and preserving the discrete symplectic structure and momentum maps. The framework specializes to the generalized rigid body on $SO(n)$, showing DEP/DLP reproducing the Moser–Veselov discretization, and offers alternative discretizations via natural charts that connect to MoV and to embedding-based approaches with constraints. The work contrasts DEP/DLP with splitting methods, highlighting the former’s exact preservation of Poisson structure and momentum and the latter’s explicitness, while also connecting to broader discrete reduction theories such as those of Bobenko and Suris. The results provide robust, momentum-preserving integrators for rigid-body and related mechanical systems on Lie groups, with clear pathways to reconstruction and extension to other manifolds.
Abstract
In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete equations provide ``reduced'' numerical algorithms which manifestly preserve the symplectic structure. The manifold $G \times G$ is used as an approximation of $TG$, and a discrete Langragian ${\mathbb L}:G \times G \to {\mathbb R}$ is construced in such a way that the $G$-invariance property is preserved. Reduction by $G$ results in new ``variational'' principle for the reduced Lagrangian $\ell:G \to {\mathbb R}$, and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when $G=\text{SO} (n)$, the DEP and DLP algorithms for a particular choice of the discrete Lagrangian ${\mathbb L}$ are equivalent to the Moser-Veselov scheme for the generalized rigid body. %As an application, a reduced symplectic integrator for two dimensional %hydrodynamics is constructed using the SU$(n)$ approximation to the volume %preserving diffeomorphism group of ${\mathbb T}^2$.