基于极式分解的$SO(n)$高阶辛李群方法
High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition
摘要：
利用极式分解和约束Galerkin方法构造了特殊正交群$SO(n)$上任意高阶的变分积分器。它的优点是避免了传统李群变分方法中指数映射的二阶导数。此外，一个减少的Lie-Poisson积分器的构造和由此产生的算法可以自然地实现的不动点迭代。在$SO(3)$上进行了数值模拟，结果表明，所提出的方法在计算效率方面与变分Runge-Kutta-Munthe-Kaas方法相当.然而，我们提出的方法更准确地保持了李群结构，并表现出更好的近能量保持。
英文摘要：
A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie--Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $SO(3)$ which demonstrate that they are comparable to variational Runge--Kutta--Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.
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