Discrete Euler-Poincaré and Lie-Poisson Equations
Jerrold E. Marsden, Sergey Pekarsky, Steve Shkoller
本文针对有限维李群G上具有G不变性的拉格朗日量L:TG→ℝ的系统，发展了Euler-Poincaré和Lie-Poisson约化理论的离散类比。这些离散方程提供了明显保持辛结构的"约化"数值算法。使用流形G×G作为TG的近似，并以保持G不变性的方式构造离散拉格朗日量𝕃:G×G→ℝ。通过G进行约化，得到了约化拉格朗日量ℓ:G→ℝ的新"变分"原理，并给出了离散Euler-Poincaré（DEP）方程。这些方程的重构恢复了<引用>中发展的离散Euler-Lagrange方程，这些方程自然是辛动量算法。此外，DEP算法的解直接导出了离散Lie-Poisson（DLP）算法。研究表明，当G=SO(n)时，对于特定选择的离散拉格朗日量𝕃，DEP和DLP算法等价于广义刚体的Moser-Veselov方案。
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 →ℝ that are G-invariant. These discrete equations provide "reduced" numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG, and a discrete Langragian 𝕃:G × G →ℝ is construced in such a way that the G-invariance property is preserved. Reduction by G results in ...
