共查询到20条相似文献，搜索用时 46 毫秒
1.
We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included. 相似文献
2.
Rodolpho Vilhena de Moraes Regina Elaine Santos Cabette Maria Cecília Zanardi Teresinha J. Stuchi Jorge Kennety Formiga 《Celestial Mechanics and Dynamical Astronomy》2009,104(4):337-353
The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient
torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed
requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined
through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical
linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized
quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian
normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s
theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases
of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability
requirements scenario. 相似文献
3.
Zsolt Sándor Bálint Érdi András Széll Barbara Funk 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):127-138
A recently introduced chaos detection method, the relative Lyapunov indicator (RLI) is investigated in the cases of symplectic mappings and of a continuous Hamiltonian system. It is shown that the RLI is an efficient and easy-to-calculate numerical tool in determining the true nature of individual orbits, and in separating ordered and regular regions of the phase space of dynamical systems. An application of the RLI for stability investigations of some recently discovered exoplanetary systems is also presented. 相似文献
4.
《Chinese Astronomy and Astrophysics》2005,29(1):81-91
In an autonomous Hamiltonian system, one constraint always exists, namely the energy integral or a constant magnitude of the 4-velocity in relativistic dynamics. The constraint should bring better numerical stability if it can be kept at every step of the numerical integration. In Newtonian mechanics, the order of the equations of motion can not be reduced by use of the constraint in most cases, because its kinetic energy is usually a quadratic form of the elliptic type and one would meet difficulty when trying the order reducing. However, the metric in general relativity is hyperbolic. In particular, when the spacetime bears some symmetries there exists a global transformation so that at least one element in its main diagonal vanishes. As a result, the constraint can be solved for a certain velocity or a momentum without any difficulty, and so reducing the order. Similarly, this technique can also be applied to the evolution of the Mixmaster universe. It is shown that this technique can raise the precision and improve the numerical stability dramatically even when a classical integrator is used, although it might not keep the symplectic structure of the system. 相似文献
5.
辛积分器中沿迹误差的一种补偿方法 总被引：2，自引：0，他引：2
辛积分器严格描述了一摄动Ｈａｍｉｌｔｏｎ系统的流，因而导致天体轨道的沿迹误差随时间呈线性增长趋势。本文利用这一特点，提出了一种对其沿迹误差进行估算的数值方法，从而达到了对数值结果进行沿迹误差补偿的目的，数值结果证实了此方法在较大积分步长和较长积分时间的数值计算中是有效的。 相似文献
6.
Alessandro Morbidelli Claude Froeschlé 《Celestial Mechanics and Dynamical Astronomy》1995,63(2):227-239
On the basis of the general theory of Hamiltonian systems, we consider the relationship between Lyapunov times and macroscopic diffusion times. We find out that there are two regimes: the Nekhoroshev regime and the resonant overlapping regime. In the first case the diffusion time is exponentially long with respect to Lyapunov times. In the second case, the relationship is polynomial although we do not find any theoretical reason for the existence of a universal power law. We show numerical evidences which confirm our theoretical considerations. 相似文献
7.
Claude Froeschlé Elena Lega Massimiliano Guzzo 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):141-153
In a previous work [Guzzo et al. DCDS B 5, 687–698 (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web. 相似文献
8.
约束条件和数值积分 总被引：3，自引：1，他引：2
自治的哈密顿系统存在约束条件，例如能量积分或广义相对论中的4速度大小为常数，它能否在数值积分过程中始终满足将直接影响数值稳定性．在牛顿力学中哈密顿系统的动能一般为椭圆型，直接运用约束条件对方程进行降阶存在开平方判断正负号的困难，导致应用高精度的经典数值积分器时能量存在耗散．然而相对论力学的度规为双曲型，利用约束条件有可能实行方程降阶．在时空具有一定对称性的情况下，能够找到整个时空的一个全局变换使变换后的度规的主对角线某一元素为零，于是从约束方程中不需开平方能够解出某一动量，顺利实现运动方程的降阶．相对论力学中另一个可以降阶的模型是Mixmaster宇宙模型．数值实验表明将经典算法用于降阶后的运动方程能够严格地满足约束，但不一定能保持辛结构。 相似文献
9.
This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and coloured noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling. 相似文献
10.
Pierre Magnenat 《Celestial Mechanics and Dynamical Astronomy》1985,35(4):329-342
Three different numerical techniques are tested to determine the number of integrals of motion in dynamical systems with three degrees of freedom.
- The computation of the whole set of Lyapunov Characteristic Exponents (LCE).
- The triple sections in the configurations space.
- The Stine-Noid box-counting technique.
11.
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
12.
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
13.
We consider Sundman and Poincaré transformations for the long-time numerical integration of Hamiltonian systems whose evolution occurs at different time scales. The transformed systems are numerically integrated using explicit symplectic methods. The schemes we consider are explicit symplectic methods with adaptive time steps and they generalise other methods from the literature, while exhibiting a high performance. The Sundman transformation can also be used on non-Hamiltonian systems while the Poincaré transformation can be used, in some cases, with more efficient symplectic integrators. The performance of both transformations with different symplectic methods is analysed on several numerical examples. 相似文献
14.
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
15.
Sergei Nayakshin Seung-Hoon Cha Alexander Hobbs 《Monthly notices of the Royal Astronomical Society》2009,397(3):1314-1325
We present a new framework for radiation hydrodynamics simulations. Gas dynamics is modelled by smoothed particle hydrodynamics (SPH), whereas radiation transfer is simulated via a time-dependent Monte Carlo approach that traces photon packets. As a first step in the development of the method, in this paper we consider the momentum transfer between radiation field and gas, which is important for systems where radiation pressure is high. There is no fundamental limitation on the number of radiation sources, the geometry or the optical depth of the problems that can be studied with the method. However, as expected for any Monte Carlo transfer scheme, stochastic noise presents a serious limitation. We present a number of tests that show that the errors of the method can be estimated accurately by considering Poisson noise fluctuations in the number of photon packets that SPH particles interact with per dynamical time. It is found that, for a reasonable accuracy, the momentum carried by photon packets must be much smaller than the typical momentum of SPH particles. We discuss numerical limitations of the code, and future steps that can be taken to improve performance and applicability of the method. 相似文献
16.
Alessandro Fortunati Stephen Wiggins 《Celestial Mechanics and Dynamical Astronomy》2016,125(2):247-262
The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called “geometric part” is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems. 相似文献
17.
《Chinese Astronomy and Astrophysics》2007,31(2):172-186
In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear. 相似文献
18.
For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the
first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian
but rather different numerical stability. When the computation of the main part of the Hamiltonian, H
0, is not expensive, we recommend to use S
* type algorithm, which cuts the calculation of the H
0 system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
J. E. Howard 《Celestial Mechanics and Dynamical Astronomy》2005,92(1-3):219-241
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. 相似文献
20.
In a previous paper (Gayon and Bois 2008a), we have shown the general efficiency of retrograde resonances for stabilizing
compact planetary systems. Such retrograde resonances can be found when two-planets of a three-body planetary system are both
in mean motion resonance and revolve in opposite directions. For a particular two-planet system, we have also obtained a new
orbital fit involving such a counter-revolving configuration and consistent with the observational data. In the present paper,
we analytically investigate the three-body problem in this particular case of retrograde resonances. We therefore define a
new set of canonical variables allowing to express correctly the resonance angles and obtain the Hamiltonian of a system harboring
planets revolving in opposite directions. The acquiring of an analytical “rail” may notably contribute to a deeper understanding
of our numerical investigation and provides the major structures related to the stability properties. A comparison between
our analytical and numerical results is also carried out. 相似文献