离散欧拉-庞加莱与李-泊松方程
Discrete Euler-Poincaré and Lie-Poisson Equations
AI总结：
本文为有限维李群上的不变系统发展了离散欧拉-庞加莱和李-泊松约化理论，导出保持辛结构的数值算法，并证明在SO(n)情形下等价于广义刚体的Moser-Veselov格式。
摘要：
本文针对有限维李群 $G$ 上具有 $G$-不变拉格朗日函数 $L:TG \to {\mathbb R}$ 的系统，发展了欧拉-庞加莱和李-泊松约化理论的离散模拟。这些离散方程提供了“约化的”数值算法，能够明显保持辛结构。流形 $G \times G$ 被用作 $TG$ 的近似，并构造离散拉格朗日函数 ${\mathbb L}:G \times G \to {\mathbb R}$，使得 $G$-不变性得以保留。通过 $G$ 的约化，为约化拉格朗日函数 $\ell:G \to {\mathbb R}$ 导出了新的“变分”原理，并提供了离散欧拉-庞加莱（DEP）方程。这些方程的重构恢复了在文献[MPS,WM]中发展的离散欧拉-拉格朗日方程，这些方程天然是辛-动量算法。此外，DEP算法的解直接导致离散李-泊松（DLP）算法。文中证明，当 $G=\text{SO} (n)$ 时，对于离散拉格朗日函数 ${\mathbb L}$ 的特定选择，DEP和DLP算法等价于广义刚体的Moser-Veselov格式。
英文摘要：
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 ${\mathbb T}^2$.
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