Explicit symplectic integrators have been important tools for accurate and
efficient approximations of mechanical systems with separable Hamiltonians. For
the first time, the article proposes for arbitrary Hamiltonians similar
integrators, which are explicit, of any even order, symplectic in an extended
phase space, and with pleasant long time properties. They are based on a
mechanical restraint that binds two copies of phase space together. Using
backward error analysis, KAM theory, and additional multiscale analysis, an
error bound of $\mathcal{O}(T\delta^l \omega)$ is established for integrable
systems, where $T$, $\delta$, $l$ and $\omega$ are respectively the (long)
simulation time, step size, integrator order, and some binding constant. For
non-integrable systems with positive Lyapunov exponents, such an error bound is
generally impossible, but satisfactory statistical behaviors were observed in a
numerical experiment with a nonlinear Schr\"{o}dinger equation.

该研究论文提出了一种基于机械约束的任意 Hamiltonians 的显式、用于扩展相空间的辛积分器，采用 KAM 理论、反向误差分析和多尺度分析建立了误差界限，对于积分系统能够达到数值统计行为的满意水平，即使在可积系统失效的情况下，也取得了较好的实验结果。

非可分哈密顿量的显式辛近似：算法与长时间性能

Explicit symplectic approximation of nonseparable Hamiltonians:  algorithm and long time performance

REBOUND is a new multi-purpose N-body code which is freely available under an
open-source license. It was designed for collisional dynamics such as planetary
rings but can also solve the classical N-body problem. It is highly modular and
can be customized easily to work on a wide variety of different problems in
astrophysics and beyond.
REBOUND comes with three symplectic integrators: leap-frog, the symplectic
epicycle integrator (SEI) and a Wisdom-Holman mapping (WH). It supports open,
periodic and shearing-sheet boundary conditions. REBOUND can use a Barnes-Hut
tree to calculate both self-gravity and collisions. These modules are fully
parallelized with MPI as well as OpenMP. The former makes use of a static
domain decomposition and a distributed essential tree. Two new collision
detection modules based on a plane-sweep algorithm are also implemented. The
performance of the plane-sweep algorithm is superior to a tree code for
simulations in which one dimension is much longer than the other two and in
simulations which are quasi-two dimensional with less than one million
particles.
In this work, we discuss the different algorithms implemented in REBOUND, the
philosophy behind the code's structure as well as implementation specific
details of the different modules. We present results of accuracy and scaling
tests which show that the code can run efficiently on both desktop machines and
large computing clusters.

REBOUND 是一个新的多功能 N 体代码，旨在解决行星环等碰撞动力学问题，并支持经典 N 体问题。它有三个辛普勒积分器，支持多种问题，在天体物理学以及其他领域中易于定制。同时，它支持多种边界条件，并可以使用 Barnes-Hut 树计算自引力和碰撞。此外，它还实现了两个基于平面扫描算法的碰撞检测模块，性能优于传统树状算法模拟。作者对 REBOUND 算法实现细节、结构以及性能进行了详细论述和测试验证。