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1.
Brett Gladman Martin Duncan Jeff Candy 《Celestial Mechanics and Dynamical Astronomy》1991,52(3):221-240
We discuss the use of symplectic integration algorithms in long-term integrations in the field of celestial mechanics. The methods' advantages and disadvantages (with respect to more common integration methods) are discussed. The numerical performance of the algorithms is evaluated using the 2-body and circular restricted 3-body problems. Symplectic integration methods have the advantages of linear phase error growth in the 2-body problem (unlike most other methods), good conservation of the integrals of the motion, good performance for moderately eccentric orbits, and ease of use. Its disadvantages include a relatively large number of force evaluations and an inability to continuously vary the step size. 相似文献
2.
The parameters of L matrices are applied to the numerical integration of regular equations describing the motion of minor bodies in the Solar System. The problem of the optimal choice of the regularizing change of variables is formulated in the context of the numerical integration of the equations of motion using the Runge–Kutta–Fehlberg method. Arbitrary perturbations are taken into account. This problem is completely solved in the case of planar motion. The solution of the optimization problem reduces the amount of computations needed to determine the vector of perturbing accelerations. Results of numerical integrations are given. 相似文献
3.
Garfinkel's solution of the Ideal Resonance problem derived from a Bohlin-von Zeipel procedure, and Jupp's solution, using Poincaré's action and angle variables and an application of Lie series expansions, are compared. Two specific Hamiltonians are chosen for the comparison and both solutions are compared with the numerical solutions obtained from direct integrations of the equations of motion. It is found that in deep resonance the second-mentioned solution is generally more accurate, while in the classical limit the first solution gives excellent agreement with the numerical integrations.This article represents a summary of a much more extensive programme of research, the complete results of which will be published in a future article. 相似文献
4.
We continue to study the number of isolating integrals in dynamical systems with three and four degrees of freedom, using as models the measure preserving mappingsT already introduced in preceding papers (Froeschlé, 1973; Froeschlé and Scheidecker, 1973a).Thus, we use here a new numerical method which enables us to take as indicator of stochasticity the variation withn of the two (respectively three) largest eigenvalues-in absolute magnitude-of the linear tangential mappingT
n
* ofT
n
. This variation appears to be a very good tool for studying the diffusion process which occurs during the disappearance of the isolating integrals, already shown in a previous paper (Froeschlé, 1971). In the case of systems with three degrees of freedom, we define and give an estimation of the diffusion time, and show that the gambler's ruin model is an approximation of this diffusion process. 相似文献
5.
This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees
of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double
oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed
by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions.
The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy
of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space
of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical
integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented
here conservation of the energy constant between 10-9 and 10-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the
number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions
were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter
any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the
boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that
investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas
of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of
sections.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Seppo Mikkola 《Celestial Mechanics and Dynamical Astronomy》1998,68(3):249-255
The inclusion of non-canonical perturbations in symplectic integration schemes has been discussed. A rigorous derivation of
an analog for theWisdom–Holman (1991) method, such that velocity dependent forces can be included, has been outlined. This
is done both by using the δfunction formalism and also by means of formal Hamiltonization. Application to the relativistic
corrections in Solar System integrations is discussed as an example. Numerical experiments confirm the usefulness of the method.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a Kepler-equation. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians. 相似文献
8.
Florent Deleflie Gilles Métris Pierre Exertier 《Celestial Mechanics and Dynamical Astronomy》2006,94(1):105-134
This paper presents an analytic solution of the equations of motion of an artificial satellite, obtained using non singular
elements for eccentricity. The satellite is under the influence of the gravity field of a central body, expanded in spherical
harmonics up to an arbitrary degree and order. We discuss in details the solution we give for the components of the eccentricity
vector. For each element, we have divided the Lagrange equations into two parts: the first part is integrated exactly, and
the second part is integrated with a perturbation method. The complete solution is the sum of the so-called “main” solution
and of the so-called “complementary” solution. To test the accuracy of our method, we compare it to numerical integration
and to the method developed in Kaula (Theory of Satellite Geodesy, Blaisdell publ. Co., New York. 1966), expressed in classical
orbital elements. For eccentricities which are not very small, the two analytical methods are almost equivalent. For low eccentricities,
our method is much more accurate. 相似文献
9.
Toshio Fukushima 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):231-241
This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are
specially designed for parallel, vector, and/or multi-processor-unit(PU) computers. The first is the Picard-Chebyshev method
(Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying
the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around
100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima,
1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules
covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method
is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with
more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial
integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by
the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs.
Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving
implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric
multistep methods (Fukushima, 1998), (Fukushima, 1999).
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.Research Assistant for the Belgian National Fund for Scientific Research 相似文献
11.
R.R. CORDEIRO A.L.F CANOVA R. VIEIRA MARTINS 《Celestial Mechanics and Dynamical Astronomy》1997,67(3):215-224
A procedure to compute the algebraic expression for eigenvectors using algebraic manipulators associated with numerical checks
is presented. This method is applied to the computation of the eigenvectors of the matrices J·D2H for the general problems with two and three degrees of freedom. Furthermore, it is used to calculate the eigenvalues‘ signature
and to analyze stability at some equilibrium points of a generalized Hénon-Heille's Hamiltonian by Krein theory.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
Marc Fouchard Christiane Froeschlé John J. Matese Giovanni Valsecchi 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):229-262
Different models of the action of the galactic tide are compared. Each model is a substitute for direct numerical integrations
allowing a drastic decrease of the computation time. The models are built using two different techniques, (i) averaging of
the fast variable (the mean anomaly) over one cometary period and (ii) fixing the comet in its aphelion direction. Moreover,
we consider two different formalisms (Lagrangian and Hamiltonian) and also two different sets of variables. As expected, we
find that the model results are independent of the formalism and the set of variables considered, and are highly accurate,
whereas mathematical technique leads to poor results. In order to further reduce the computation time, mappings are built
from the development of the solution of the models. We show that for these mappings, the set of variables giving the most
accurate results is strongly dependent on the cometary eccentricity, e, and semimajor axis, a. 相似文献
13.
P. K. Seidelmann 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):1-12
Planetary and satellite theories have been historically and are presently intimately related to the available computing capabilities, the accuracy of observational data, and the requirements of the astronomical community. Thus, the development of computers made it possible to replace planetary and lunar general theories with numerical integrations, or special perturbation methods. In turn, the availability of inexpensive small computers and high-speed computers with inexpensive memory stimulated the requirement to change from numerical integration back to general theories, or representative ephemerides, where the ephemerides could be calculated for a given date rather than using a table look-up process. In parallel with this progression, the observational accuracy has improved such that general theories cannot presently achieve the accuracy of the observations, and, in turn, it appears that in some cases the models and methods of numerical integration also need to be improved for the accuracies of the observations. Planetary and lunar theories were originally developed to be able to predict phenomena, and provide what are now considered low accuracy ephemerides of the bodies. This proceeded to the requirement for high accuracy ephemerides, and the progression of accuracy improvement has led to the discoveries of the variable rotation of the Earth, several planets, and a satellite. By means of mapping techniques, it is now possible to integrate a model of the motion of the entire solar system back for the history of the solar system. The challenges for the future are: Can general planetary and lunar theories with an acceptable number of terms achieve the accuracies of observations? How can numerical integrations more accurately represent the true motions of the solar system? Can regularly available observations be improved in accuracy? What are the meanings and interpretations of stability and chaos with respect to the motions of the bodies of our solar system? There has been a parallel progress and development of problems in dealing with the motions of artificial satellites. The large number of bodies of various sizes in the limited space around the Earth, subject to the additional forces of drag, radiation pressure, and Earth zonal and tesseral forces, require more accurate theories, improved observational accuracies, and improved prediction capabilities, so that potential collisions may be avoided. This must be accomplished by efficient use of computer capabilities. 相似文献
14.
Marc A. Murison 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):175-180
The restricted three-body problem (RTBP) has in the past played an essential role in many different areas of dynamical astronomy, and indications are that this will continue. As the state of the art in computing becomes more advanced, larger numbers of integrations and longer durations are attempted. Thus, computational efficiency and accuracy are becoming more important. Also, the use of the RTBP in many different areas leads to the desire for a general integration method.In order to maximize the efficiency of orbit calculations, comparisons are made of different methods of integration. The results can be summarized as follows: 1. The Bulirsch-Stoer extrapolation method is extremely fast and accurate, and is the method of choice. 2. Regularization of the equations of motion is essential. 3. When applicable, a manifold correction algorithm, originally due to Nacozy (1971), reduces numerical errors to the limits of machine accuracy, and at a cost of only 1 to 3 percent in cpu time. 相似文献
15.
Nadia A. Saad 《Astrophysics and Space Science》2003,284(3):1001-1012
The stellar problem of three bodies is studied. A Lie transform is used to eliminate the short and intermediate periodic terms
from Harrington's Hamiltonian. The resulting system is a one of two degrees of freedom whose further reduction requires recourse
to numerical procedures.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
16.
Michael E. Hough 《Celestial Mechanics and Dynamical Astronomy》1995,62(3):263-287
A new integration theory is formulated for dynamical systems with two degrees of freedom, in the gravitational field of a rotating system. Four integrals of motion may be determined from complete solutions of a system of three first-order, partial differential equations in three independent variables. The solutions of this system define two integral surfaces with space-time coordinates. These surfaces represent two independent solutions of a second-order kinematic system to which the original fourth-order system has been reduced. An integral curve may be represented as the locus of intersection points of the integral surfaces. The new theory is the theoretical basis for a method of analytic continuation of periodic orbits of the circular restricted problem. 相似文献
17.
A new scale transformation to the integrated velocity vector is designed to monitor the accumulation of numerical errors in
several integrals of motion. The scale factor is derived from the least-squares correction that minimizes the sum of the squares
of the errors of these integrals. In order to preserve an invariant, we employ the velocity scaling method for rigorously
satisfying the constraint. When adjusting many constants, the new scheme like other existing methods is valid to typically
reduce the integration errors below those of an uncorrected integrator. Via integral invariant relations, the new method is
also able to treat slowly-varying quantities, such as the Keplerian energy and the Laplace vector, for a perturbed Keplerian
problem or each of multiple bodies in the solar system dynamics. Consequently it does nearly agree with the rigorous dual
scaling method in the sense of drastically improving the integration accuracy. As one of its advantages, the implementation
of the new method is significantly easier than that of other methods. In particular, the method can be simply applied to a
complicated dynamical system with some constraints. 相似文献
18.
We present statistical distributions of Jovian perturbations on short-period comet orbits resulting from accurate numerical integrations. Our sample of 60, 000 cometary orbits with low inclinations and random orientations is characterized by perihelia between 0 and 7 AU and aphelia between 4 and 13 AU. The perturbations considered are those experienced because of Jupiter's gravitation per orbital revolution by the comets. Regularization and accurate step-length control in the numerical integration gives statistical results appreciably different from those computed by Rickman and Vaghi (1978). Their use of a crude method of integration led to erroneous results for close encounters. Strong asymmetries of the distributions, in particular for the extreme tails, are observed for perihelion- or aphelion-tangent orbits. These orbits are also shown to experience the strongest energy perturbations on the average. Some results concerning the perturbations of Tisserand parameters are indicated. The perturbation distributions for the angular elements are described and discussed. The role of the minimum distance from Jupiter as an indicator of perturbations is investigated. 相似文献
19.
The 221-resonance case for a potential problem with three degrees of freedom is characterized by the existence of two isolating approximate integrals apart from the energy. This result completes a statement by Gustavson concerning the number of formal integrals in resonant Hamiltonian systems. 相似文献