辛积分器的物理学：近日点进展和辛校正算法
The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms
摘要：
辛积分器根据修正的哈密顿量来演化动力系统，修正的哈密顿量的误差项也是定义良好的哈密顿量。该算法的误差是每个误差哈密顿量对精确解的扰动之和。当辛积分应用于开普勒问题时，这些误差项会导致轨道进动。在这项工作中，通过开发一个通用的方法计算近日点推进通过拉普拉斯-龙格-伦茨矢量，即使是不可分的哈密顿，我表明，在辛积分的进动误差可以计算分析。发现在每一阶，每一对误差哈密顿量使轨道在每一个周期后以完全相同的量反向进动。因此，对于这些成对误差项具有相等系数的辛校正器或过程积分器，将在每个周期之后精确地消除其进动误差。因此，辛积分器的物理性质决定了积分长时间周期运动的最佳算法。
英文摘要：
Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for non-separable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at each order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors exactly cancel after each period. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions.
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