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1.
A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H=A+B was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(p + 22), where is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(p +42). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn. 相似文献
2.
Hiroshi Kinoshita Haruo Yoshida Hiroshi Nakai 《Celestial Mechanics and Dynamical Astronomy》1990,50(1):59-71
Symplectic integrators have many merits compared with traditional integrators: 相似文献
3.
Taeyoung Lee Melvin Leok N. Harris McClamroch 《Celestial Mechanics and Dynamical Astronomy》2007,98(2):121-144
Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under
their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s
principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion,
referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method
for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good
energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the
gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without
the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell
bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational
integrator compared with integrators that are not symplectic or do not preserve the Lie group structure. 相似文献
4.
We build high order numerical methods for solving differential equations by applying extrapolation techniques to a Symplectic
Integrator of order 2n. We show that, in general, the qualitative properties are preserved at least up to order 4n+1. This new procedure produces much more efficient methods than those obtained using the Yoshida composition technique.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
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
6.
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
7.
Patrick Michel Giovanni H. Valsecchi 《Celestial Mechanics and Dynamical Astronomy》1996,65(4):355-371
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. 相似文献
8.
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
9.
K.G. Hadjifotinou 《Planetary and Space Science》2002,50(4):361-369
A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J2 and J4. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems. 相似文献
10.
François Farago Jacques Laskar Jocelyn Couetdic 《Celestial Mechanics and Dynamical Astronomy》2009,104(3):291-306
Exploring the global dynamics of a planetary system involves computing integrations for an entire subset of its parameter
space. This becomes time-consuming in presence of a planet close to the central star, and in practice this planet will be
very often omitted. We derive for this problem an averaged Hamiltonian and the associated equations of motion that allow us
to include the average interaction of the fast planet. We demonstrate the application of these equations in the case of the μ Arae system where the ratio of the two fastest periods exceeds 30. In this case, the effect of the inner planet is limited
because the planet’s mass is one order of magnitude below the other planetary masses. When the inner planet is massive, considering
its averaged interaction with the rest of the system becomes even more crucial. 相似文献
11.
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
12.
Renu Malhotra 《Celestial Mechanics and Dynamical Astronomy》1994,60(3):373-385
Recently a new class of numerical integration methods — mixed variable symplectic integrators — has been introduced for studying long-term evolution in the conservative gravitational few-body problem. These integrators are an order of magnitude faster than conventional ODE integration methods. Here we present a simple modification of this method to include small non-gravitational forces. The new scheme provides a similar advantage of computational speed for a larger class of problems in Solar System dynamics. 相似文献
13.
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. 相似文献
14.
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
15.
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
16.
A new algorithm is developed for long-term integrations of the N-body problem. The method uses symplectic integrations of the Hamiltonian equations of motion for each body. This allows one
to employ individual adaptive time-steps in computations. The efficiency of this technique is demonstrated by several tests
performed for typical problems of Solar System dynamics. 相似文献
17.
Karl Wodnar 《Celestial Mechanics and Dynamical Astronomy》1996,65(1-2):85-94
This paper deals with some recent problems of frequency determination in dynamical systems. As a main result a new Fast Fourier Transform (FIT) method, the High Resolution Fast Fourier Transform (HRFFT) is presented, which allows for economic evaluation of discrete Fourier spectra at arbitrary frequencies, not just at the evenly spaced points of the rigid frequency grid the ordinary FFT is restricted to (continuous evaluation). Comparison is made to existing methods in what regards continuous evaluation, computational speed, memory requirements and precision, evidencing the high efficiency of the HRFFT. Possible applications of the new technique are indicated, showing that existing methods, as e.g. that of Laskar (1990, Icarus 88) and its extension by idlichovský and Nesvorný (1996, this volume), may strongly benefit from the HRFFT. 相似文献
18.
A quick analytical method is presented for calculating comet cloud formation efficiency in the case of a single planet or
multiple-planet system for planets that are not too eccentric (e
p
≲ 0.3). A method to calculate the fraction of comets that stay under the control of each planet is also presented, as well
as a way to determine the efficiency in different star cluster environments. The location of the planet(s) in mass-semi-major
axis space to form a comet cloud is constrained based on the conditions developed by Tremaine (1993) together with estimates
of the likelyhood of passing comets between planets; and, in the case of a single, eccentric planet, the additional constraint
that it is, by itself, able to accelerate material to relative encounter velocity U ~ 0.4 within the age of the stellar system without sweeping up the majority of the material beforehand. For a single planet,
it turns out the efficiency is mainly a function of planetary mass and semi-major axis of the planet and density of the stellar
environment. The theory has been applied to some extrasolar systems and compared to numerical simulations for both these systems
and the Solar System, as well as a diffusion scheme based on the energy kick distribution of Everhart (Astron J 73:1039–1052,
1968). The analytic results are in good agreement with the simulations. 相似文献
19.
20.
In this paper we show how the equations of motion for the smoothed particle hydrodynamics (SPH) method may be derived from a variational principle for both non-relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the general relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation. 相似文献