共查询到20条相似文献，搜索用时 15 毫秒
1.
Paul Nacozy 《Celestial Mechanics and Dynamical Astronomy》1976,13(4):495-501
A review and discussion of several investigations concerning the effect of time transformations on numerical integration errors is given. In particular, the discussion treats the relation between time transformations andlocal truncation errors. Additional numerical results are presented which indicate that time transformations reducelocal truncation errors. The results complement those of other studies, especially the recent studies of Danby, Wong, Velez, and Feagin and Mikkilineni. A Sundman time transformation with avarying exponent is introduced and discussed. 相似文献
2.
Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique. We construct piecewise polynomial maps by coarse-graining the phase surface of section into parallelograms using values of the Poincaré maps at the vertices to define a polynomial approximation within each cell. The numerical experiments are in good agreement with the standard map taken as a model problem. The agreement is better when the number of vertices and the order of the polynomial fit increase. The synthetic mapping obtained is not symplectic even if at vertices there is an exact interpolation. We introduce a second new method based on a global fitting . The polynomials are obtained using at once all the vertices and fitting by least square polynomes but in such a way that the symplectic character is not lost. 相似文献
3.
Paul E. Nacozy 《Celestial Mechanics and Dynamical Astronomy》1981,23(2):173-198
Time elements are introduced in terms of Keplerian (classical) orbital elements for use with time transformations of the Sundman type. Three different time elements are introduced. One time element is associated with the eccentric anomaly, a second time element is associated with the true anomaly, and a third time element is associated with theintermediate anomaly.Numerical results are presented that show accuracy improvements of from one to two orders of magnitude when time elements are employed along with Sundman time transformations, compared with using time transformations alone. 相似文献
4.
We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori–Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincaré system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems. 相似文献
5.
《New Astronomy》2016
We study the dynamical chaos and integrable motion in the planar circular restricted three-body problem and determine the fractal dimension of the spiral strange repeller set of non-escaping orbits at different values of mass ratio of binary bodies and of Jacobi integral of motion. We find that the spiral fractal structure of the Poincaré section leads to a spiral density distribution of particles remaining in the system. We also show that the initial exponential drop of survival probability with time is followed by the algebraic decay related to the universal algebraic statistics of Poincaré recurrences in generic symplectic maps. 相似文献
6.
We present a generalisation of the Levi-Civita and Kustaanheimo-Stiefel regularisation. This allows the use of more general time rescalings. In particular, it is possible to find a regularisation which removes the singularity of the equations and preserves scaling invariance. In addition, these equations can, in certain cases, be integrated with explicit symplectic Runge-Kutta-Nyström methods. The combination of both techniques gives an explicit adaptive symplectic (EASY) integrator. We apply those methods to some perturbations of the Kepler problem and illustrate, by means of some numerical examples, when scaling invariant regularisations are more efficient that the LC/KS regularisation. 相似文献
7.
辛算法在动力天文中的应用（Ⅲ） 总被引：3，自引：0，他引：3
文［１］和文［２］从哈密顿系统的整体结构保持一角度阐明了辛算法［３－６］的主要功能，本文将从定量的角度进一步表明辛算法的另一独特优点－可以控制天体运动沿迹误差的快速增长，并对可分离哈密顿系统的显式辛差分格式稍加改进，推广应用到一般动力系统，该系统含有小耗散项或小的不可分离项，计算结果表明，效果极佳，因此，辛算法与传统的数值解法相比，确有很多优点。 相似文献
8.
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
9.
当史瓦西黑洞周围存在渐近均匀的外部磁场时, 描述带电粒子在史瓦西黑洞附近运动的哈密顿系统会变为不可积系统. 类似于这样的相对论哈密顿系统不存在有显式分析解的2部分分离形式, 给显式辛算法的构建和应用带来困难. 近一年以来的系列工作提出将相对论哈密顿系统分解为具有显式分析解的2个以上分离部分形式, 成功解决了许多相对论时空构建显式辛算法的难题. 最近的工作回答了哈密顿系统显式可积分离数目对长期数值积分精度有何影响、哪种显式辛算法有最佳长期数值性能这两个问题, 指出哈密顿有最小可积分离数目即3部分分裂解形式并且应用于优化的4阶分段龙格库塔显式辛算法可取得最好精度. 由此选择上述数值积分方法并利用庞加莱截面、最大李雅普诺夫指数和快速李雅普诺夫指标研究在磁化史瓦西黑洞附近运动的带电粒子轨道动力学. 结果显示: 针对某特定的粒子能量和角动量, 较小的外部磁场很难形成混沌轨道; 较大的正磁场参数容易使轨道产生混沌, 并且随着磁场的增大, 轨道的混沌程度也随之加强; 粒子能量适当变大也可以加剧混沌程度, 但负磁场参数和粒子角动量变大都会减弱混沌. 相似文献
10.
Michel Rapaport 《Celestial Mechanics and Dynamical Astronomy》1976,13(2):217-227
We use classical definitions and results of differential geometry in studying properties of transformations depending on a small parameter, acting on differential systems. Hori's and Deprit's algorithms can be defined for these systems. A lemma is given to show these algorithms are equivalent. The so-called property of covariance is merely established. The canonical systems are then considered as associated with Hamiltonian vectorfields on symplectic manifolds. The property that the infinitesimal generator of a canonical transformation is an Hamiltonian vectorfield permits to establish separately the generality of Hori's and Deprit's algorithms. We suggest that the Hamiltonian vectorfield property can be extended to the generators of transformations depending on several parameters. 相似文献
11.
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
12.
Haruo Yoshida 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):27-43
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. 相似文献
13.
In a paper by the second author (Nacozy, 1981), various time elements are presented for use with the Sundman time transformation. In that paper, the time elements are given in terms of Keplerianorbital elements. We give here the corresponding time elements in terms ofrectangular coordinates. Extensive references are given in the previous paper and will be omitted here.We present additional numerical experiments comparing the use of time elementsand time transformationstogether with the use of time transformationsalone. The results indicate a reduction in computational error when time elements are used. 相似文献
14.
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
15.
Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small
parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on
Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same
functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition
to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving
power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent
possible.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
16.
We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the Pythagorean orbit, and a periodic collision orbit. 相似文献
17.
Seppo Mikkola 《Celestial Mechanics and Dynamical Astronomy》1997,67(2):145-165
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. 相似文献
18.
P. Stumpff 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):211-222
My father K. Stumpff (1947, 1949, 1951, 1959, 1962) developed a transcendental equation which replaces the original Kepler equation but is valid for all types of orbits. Other advantages over the classical methods are: a) the independent arguments of the equation follow from the vectors of position and velocity at any instant To, where To is not necessarily the perihelion time; b) an explicit knowledge of the classical orbital elements is not required; c) transformations of coordinate systems are avoided. The present paper discusses the properties of the general Kepler equation in a wide range of its independent arguments, and it is shown that analytic solutions, existing in special cases, can be used for the numerical solution of general cases. The theory is generalized insofar as it now can handle not only attracting forces but also repulsive ones. As a result of this investigation, FORTRAN subroutines were developed which can be used in connection with any two-body problem for the computation of position and velocity as function of time along any unperturbed orbit. 相似文献
19.
J. M. Beckers 《Solar physics》1968,5(1):15-28
The principles of operation of photoelectric solar magnetographs are described in terms of the Poincaré sphere. The performance of photographic methods for measuring solar magnetic fields is compared with that of photoelectric magnetographs. 相似文献
20.
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