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1.
Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in
the N-body problem. These forms are introduced, compared to previous approaches, and their properties are discussed. Applications
to symplectic mappings, the definition of action-angle variables for the Kepler problem, and Hamiltonian perturbation theory
are outlined
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
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. 相似文献
3.
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. 相似文献
4.
Haruo Yoshida 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):355-364
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. 相似文献
5.
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. 相似文献
6.
A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiter's Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn. 相似文献
7.
Seppo Mikkola 《Celestial Mechanics and Dynamical Astronomy》1999,74(4):275-285
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. 相似文献
8.
Vacheslav Emel'yanenko 《Celestial Mechanics and Dynamical Astronomy》2002,84(4):331-341
A new symplectic algorithm is developed for cometary orbit integrations. The integrator can handle both high-eccentricity orbits and close encounters with planets. The method is based on time transformations for Hamiltonians separated into Keplerian and perturbation parts. The adaptive time-step of this algorithm depends on the distance from a centre and the magnitude of perturbations. The explicit leapfrog technique is simple and efficient. 相似文献
9.
Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic
leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and
the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures
from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in
general.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
Processing techniques are used to approximate the exact flow of near-integrable Hamiltonian systems depending on a small perturbation
parameter. We study the reduction of the number of conditions for the kernel for this type of Hamiltonians and we build third,
fourth and fifth order methods which are shown to be more efficient than previous algorithms for the same class of problems.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
12.
13.
Zsolt Sándor Bálint Érdi Carl D. Murray 《Celestial Mechanics and Dynamical Astronomy》2002,84(4):355-368
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. 相似文献
14.
Massimiliano Guzzo 《Celestial Mechanics and Dynamical Astronomy》2001,80(1):63-80
I have improved the precision of the leap–frog symplectic integrators for perturbed Kepler problems at small eccentricities, without significant loss of CPU time. The integration scheme proposed is competitive, in some situations, with the so-called mixed variable integrators. 相似文献
15.
几类辛方法的数值稳定性研究 总被引:1,自引:0,他引:1
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. 相似文献
16.
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. 相似文献
17.
We deal with the problem of a zero mass body oscillating perpendicular to a plane in which two heavy bodies of equal mass orbit each other on Keplerian ellipses. The zero mass body intersects the primaries plane at the systems barycenter. This problem is commonly known as theSitnikov Problem. In this work we are looking for a first integral related to the oscillatory motion of the zero mass body. This is done by first expressing the equation of motion by a second order polynomial differential equation using a Chebyshev approximation techniques. Next we search for an autonomous mapping of the canonical variables over one period of the primaries. For that we discretize the time dependent coefficient functions in a certain number of Dirac Delta Functions and we concatenate the elementary mappings related to the single Delta Function Pulses. Finally for the so obtained polynomial mapping we look for an integral also in polynomial form. The invariant curves in the two dimensional phase space of the canonical variables are investigated as function of the primaries eccentricity and their initial phase. In addition we present a detailed analysis of the linearized Sitnikov Problem which is valid for infinitesimally small oscillation amplitudes of the zero mass body. All computations are performed automatically by the FORTRAN program SALOME which has been designed for stability considerations in high energy particle accelerators. 相似文献
18.
Sławomir Breiter 《Celestial Mechanics and Dynamical Astronomy》1998,71(4):229-241
An explicit symplectic integrator is constructed for perturbed elliptic orbits of an arbitrary eccentricity. The perturbation
should be Hamiltonian, but it may depend on time explicitly. The main feature of the integrator is the use of KS variables
in the ten-dimensional extended phase space. As an example of its application the motion of an Earth satellite under the action
of the planet's oblateness and of lunar perturbations is studied. The results confirm the superiority of the method over a
classical Wisdom–Holman algorithm in both accuracy and computation time.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
We present a new method for fast numerical integration of close binaries inN-body systems. The basic idea is to slow down the motion of the binary artificially, which makes a faster numerical integration possible but still maintains correct treatment of secular and long-period effects on the motion. We discuss the general principle, with application to close binaries inN-body codes and in the chain regularization. 相似文献
20.
Xinhao Liao 《Celestial Mechanics and Dynamical Astronomy》1996,66(3):243-253
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. 相似文献