Recent progress in the theory and application of symplectic integrators

In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.     

辛方法的校正公式

1996年Wisdom等提出了对辛方法进行校正的概念和实践，现在继续对辛校正进行详尽讨论和数值比较，尤其对哈密顿函数可分解为一个主要部分和多个次要部分的一般情形，用Lie级数推导任意阶的各种辛算法的一次和二次辛校正公式并对一些算法给出具体的辛校正公式。又以日、木、土三体问题为模型进行数值实验，结果表明一次辛校正能提高精度，改善数值稳定性。计算效率也比较高，因而值得推荐使用，辛方法通常用大步长数值积分，这时二次辛校正并没有显著提高结果的精度，却大大增加了计算时间，不应予以推荐。     

Forward Symplectic Integrators for Solving Gravitational Few-Body Problems

We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in addition to the force. We demonstrate the efficiency of these Forward Symplectic Integrators by solving the circular restricted three-body problem in the space-fixed frame where the force on the third body is explicitly time-dependent. These algorithms can achieve accuracy of Runge–Kutta, conventional negative time step symplectic and corrector symplectic algorithms at step sizes five to ten times as large.     

Singularly weighted symplectic forms and applications to asteroid motion

New techniques to study Hamiltonian systems with Hamiltonian forcing are proposed. They are based on singularly weighted symplectic forms and transformations which preserve these forms. Applications pertaining to asteroid motion are outlined. These involve the presence of both Jupiter and Saturn.     

Symplectic Mappings of Co-orbital Motion in the Restricted Problem of Three Bodies

The dynamics of co-orbital motion in the restricted three-body problem are investigated by symplectic mappings. Analytical and semi-numerical mappings have been developed and studied in detail. The mappings have been tested by numerical integration of the equations of motion. These mappings have been proved to be useful for a quick determination of the phase space structure reflecting the main characteristics of the dynamics of the co-orbital problem.     

Explicit Symplectic Algorithms For Time‐Transformed Hamiltonians

By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space. Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator. In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic N‐body integration. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical experiments on the efficiency of second-order mixed-variable symplectic integrators for N-body problems

We discuss the efficiency of the so-called mixed-variable symplectic integrators for N-body problems. By performing numerical experiments, we first show that the evolution of the mean error in action-like variables is strongly dependent on the initial configuration of the system. Then we study the effect of changing the stepsize when dealing with problems including close encounters between a particle and a planet. Considering a previous study of the slow encounter between comet P/Oterma and Jupiter, we show that the overall orbital patterns can be reproduced, but this depends on the chosen value of the maximum integration stepsize. Moreover the Jacobi constant in a restricted three-body problem is not conserved anymore when the stepsize is changed frequently: over a 10⁵ year time span, to keep a relative error in this integral of motion of the same order as that given by a Bulirsch-Stoer integrator requires a very small integration stepsize and much more computing time. However, an integration of a sample including 10⁴ particles close to Neptune shows that the distributions of the variation of the elements over one orbital period of the particles obtained by the Bulirsch-Stoer integrator and the symplectic integrator up to a certain integration stepsize are rather similar. Therefore, mixed-variable symplectic integrators are efficient either for N-body problems which do not include close encounters or for statistical investigations on a big sample of particles.     

Symplectic Integrators: Rotations and Roundoff Errors

We investigate the numerical implementation of a symplectic integrator combined with a rotation (as in the case of an elongated rotating primary). We show that a straightforward implementation of the rotation as a matrix multiplication destroys the conservative property of the global integrator, due to roundoff errors. According to Blank et al. (1997), there exists a KAM-like theorem for twist maps, where the angle of rotation is a function of the radius. This theorem proves the existence of invariant tori which confine the orbit and prevent shifts in radius. We replace the rotation by a twist map or a combination of shears that display the same kind of behaviour and show that we are able not only to recover the conservative properties of the rotation, but also make it more efficient in term of computing time. Next we test the shear combination together with symplectic integrator of order 2, 4, and 6 on a Keplerian orbit. The resulting integrator is conservative down to the roundoff errors. No linear drift of the energy remains, only a divergence as the square root of the number of iterations is to be seen, as in a random walk. We finally test the three symplectic integrators on a real case problem of the orbit of a satellite around an elongated irregular fast rotating primary. We compare these integrators to the well-known general purpose, self-adaptative Bulirsch–Stoer integrator. The sixth order symplectic integrator is more accurate and faster than the Bulirsch–Stoer integrator. The second- and fourth- order integrators are faster, but of interest only when extreme speed is mandatory. This revised version was published online in August 2006 with corrections to the Cover Date.     

Non-Existence of the Modified First Integral by Symplectic Integration Methods II: Kepler Problem

Symplectic integration methods conserve the Hamiltonian quite well because of the existence of the modified Hamiltonian as a formal conserved quantity. For a first integral of a given Hamiltonian system, the modified first integral is defined to be a formal first integral for the modified Hamiltonian. It is shown that the Runge-Lenz vector of the Kepler problem is not well conserved by symplectic methods, and that the corresponding modified first integral does not exist. This conclusion is given for a one-parameter family of symplectic methods including the symplectic Euler method and the Störmer/Verlet method.     

Explicit Symplectic Integrator for Rotating Satellites

An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach: the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator proves to be quite efficient. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence of formal integrals of symplectic integrators

A recurrent method of solving the formal integrals of symplectic integrators is given. The special examples show that there are no long-term variations in all integrals of the Hamiltonian system in addition to the energy one when symplectic integrators are used in the numerical studies of the system. As an application of the formal integrals, the relation between them and the linear stability of symplectic integrators is discussed.     

Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.     

Practical Symplectic Methods with Time Transformation for the Few-Body Problem

The use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed. Numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies. Applications extend from perturbed two-body motion to hierarchical many-body systems with large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regularizing Time Transformations in Symplectic and Composite Integration

We consider numerical integration of nearly integrable Hamiltonian systems. The emphasis is on perturbed Keplerian motion, such as certain cases of the problem of two fixed centres and the restricted three-body problem. We show that the presently known methods have useful generalizations which are explicit and have a variable physical timestep which adjusts to both the central and perturbing potentials. These methods make it possible to compute accurately fairly close encounters. In some cases we suggest the use of composite (instead of symplectic) alternatives which typically seem to have equally good energy conservation properties.This revised version was published online in October 2005 with corrections to the Cover Date.     

关于近地小行星轨道演化的初步探索

本文采用改进的显式辛算法和嵌套的PKF7（8）积分器同时对86颗已命名（或编号）的近地小行星的轨道演化进行了数值研究，在103－104年的时间尺度上，给出了这些小行星轨道演化的状况以及它们与几颗大行星靠近的最小距离，特别是与地球接近的最小距离可小于0．01天文单位，甚至可能比月球还更靠近地球．     

Efficient Symplectic Integration of Satellite Orbits

The use of the extended phase space and time transformations for constructing efficient symplectic methods for computing the long term behavior of perturbed two‐body systems are discussed. Main applications are for artificial satellite orbits. The methods suggested here are efficient also for large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simple derivation of Symplectic Integrators with First Order Correctors

In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(h ⁶)+(² h ²), where is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(h ⁶)+(² h ⁴).This revised version was published online in October 2005 with corrections to the Cover Date.     

几类辛方法的数值稳定性研究

主要对一阶隐式Euler辛方法M1、二阶隐式Euler中点辛方法M2、一阶显辛Euler方法M3和二阶leapfrog显辛积分器M4共4种辛方法及一些组合算法进行了通常意义下的线性稳定性分析．针对线性哈密顿系统,理论上找到每个数值方法的稳定区,然后用数值方法检验其正确性．对于哈密顿函数为实对称二次型的情况,为了理论推导便利,特推荐采用相似变换将二次型的矩阵对角化来研究辛方法的线性稳定性．当哈密顿分解为一个主要部分和一个小摄动次要部分且二者皆可积时,无论是线性系统还是非线性系统,这种主次分解与哈密顿具有动势能分解相比,明显扩大了辛方法的稳定步长范围．     

Stability and Geometry of Third-Order Resonances in four-Dimensional Symplectic Mappings

We analyze four-dimensional symplectic mappings in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a third-order resonance. Using the perturbative tools of resonant normal forms, the geometry of the orbits and the existence of elliptic or hyperbolic one-dimensional tori (fixed lines) is worked out. This allows one to give an analytical estimate of the stability domain when the resonance is unstable. A comparison with numerical results for the four-dimensional Hénon mapping is given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Symplectic integrator for general near-integrable Hamiltonian system

In this paper, following the idea of constructing the mixed symplectic integrator (MSI) for a separable Hamiltonian system, we give a low order mixed symplectic integrator for an inseparable, but nearly integrable, Hamiltonian system, Although the difference schemes of the integrators are implicit, they not only have a small truncation error but, due to near integrability, also a faster convergence rate of iterative solution than ordinary implicit integrators, Moreover, these second order integrators are time-reversible.