共查询到20条相似文献，搜索用时 656 毫秒
1.
Charles E. Roberts Jr. 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):397-407
The author relates his experiences in utilizing the power series method to generate trajectories for orbital and sub-orbital vehicles and for then-body problem. 相似文献
2.
Juan F. Navarro 《Celestial Mechanics and Dynamical Astronomy》2018,130(2):16
The aim of this article is to present a method for the integration of the equations of motion of the N-body ring problem by means of recurrent power series. We prove that the solution is convergent for any set of initial conditions, excluding those corresponding to binary collisions. 相似文献
3.
4.
Christoph Lhotka 《Celestial Mechanics and Dynamical Astronomy》2017,127(4):397-428
In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle. 相似文献
5.
Edelman Colette 《Celestial Mechanics and Dynamical Astronomy》1993,55(4):369-386
The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS 1 andS 2 changing the eccentricity of a solution. The family of symmetries,S 1, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S 2, is related to a possibleaffinity of the configurations space. Via a symmetry of theS 2 family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS 1.S 2, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC 2 H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:
- The two families of symmetriesS 1 andS 2 are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
- TheS 1 family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
- TheS 2 family of symmetries may regularize multiple collisions among central solutions of the same type.
6.
Florin N. Diacu 《Celestial Mechanics and Dynamical Astronomy》1988,44(3):261-265
It is shown that in then-body problem with generalized attraction law, the sets of initial conditions which lead to Wintner's collinear, respectively flat, motion are nowhere dense relative to the set of initial conditions that define solutions inR
3. 相似文献
7.
R. F. Arenstorf 《Celestial Mechanics and Dynamical Astronomy》1978,17(4):331-355
Letn2 mass points with arbitrary masses move circularly on a rotating straight-line central-configuration; i.e. on a particular solution of relative equilibrium of then-body problem. Replacing one of the mass points by a close pair of mass points (with mass conservation) we show that the resultingN-body problem (N=n+1) has solutions, which are periodic in a rotating coordinate system and describe precessing nearlyelliptic motion of the binary and nearlycircular collinear motion of its center of mass and the other bodies; assuming that also the mass ratio of the binary is small. 相似文献
8.
Thierry Combot 《Celestial Mechanics and Dynamical Astronomy》2012,114(4):319-340
We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree ?1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n-body problem with Newtonian interaction in the plane on a surface of equation (H, C) = (H 0, C 0) with (H 0, C 0) ?? (0, 0) where C is the total angular momentum and H the Hamiltonian, in the case where the n masses are equal. Several other cases in the 3-body problem are also proved to be non integrable in the same way, and some examples displaying partial integrability are provided. 相似文献
9.
Robert L. Devaney 《Celestial Mechanics and Dynamical Astronomy》1979,19(4):391-404
We investigate specific homothetic solutions of then-body problem which both begin and end in a simultaneous collision of all of the particles. Under a suitable change of variables, these solutions become heteroclinic orbits, i.e., they lie in the intersection of the stable and unstable manifolds of distinct equilibrium points. Our main result is that these manifolds intersect transversely along these orbits. This proves that the homothetic solutions are structurally stable.Partially supported by NSF Grant MCS 77-00430. 相似文献
10.
W. Black 《Celestial Mechanics and Dynamical Astronomy》1973,8(3):357-370
It is shown that in the numerical integration ofN-body problems, as much importance should be given to considerations of the computer programming language to be used as to questions of the accummulation of round-off and truncation error, the stability of the method chosen and the problem being treated. By careful programming processing time may be cut by a factor of 2 or 3 which is an important consideration in extended numerical investigations. The relative usefulness of differing strategies for determining the step size is discussed and in addition the usefulness is shown of treatingN-body problems by a Taylor series method. 相似文献
11.
Carles Simó 《Celestial Mechanics and Dynamical Astronomy》1978,18(2):165-184
Beyond the casen=3 little was known about relative equilibrium solutions of then-body problem up to recent years. Palmore's work provides in the general case much useful information. In the casen=4 he gives the totality of solutions when the four masses are equal and studies some degeneracies. We present here a survey of solutions for arbitrary masses, discussing the manifolds of degeneracy. The ordering of restricted potentials allows a counting of the number of bifurcation sets and different invariant manifolds. An analysis of linear stability is done in the restricted and general cases. As a result, values of the masses ensuring linear stability are given. 相似文献
12.
Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c 2 h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL 2 collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec 2 h ‘integrals’,z 23 andz 34, att=0 are assumed to be smaller than the corresponding critical values \(\tilde z_{23} \) and \(\tilde z_{34} \) , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz 23 orz 34 is changed by an amount \(\tilde z_{ij} - z_{ij} \left( 0 \right)\) ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between \(\Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-0em} {\dot z_{23} }}\) and \(\Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-0em} {\dot z_{34} }}\) . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, the? ij of Roy and Walker. When an appropriate perturbation theory is based on the? ij , the asymptotic expansions are much more rapidly decreasing than the usual expansions in powers of the mass ratios (as in the classical Lagrange perturbation theory) and can be extended also to cases such as lunar theory or double binaries. The second step is the computation of the time derivatives \(\dot z_{ij} \) (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in the? ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of Δt 23, Δt 34. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the Δt ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez ij . The same method could be applied also toN≥5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of two? ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is ～10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics. 相似文献
13.
Richard J. Fateman 《Celestial Mechanics and Dynamical Astronomy》1974,10(2):243-247
The advantages of a tree-like data structure for use in the multiplication of Poisson series by computer are espoused. Compared to list representations, computations with trees can be faster by a factor ofn/log(n), wheren is the number of trigonometric terms in a multiplicand. 相似文献
14.
Donald G. Saari 《Celestial Mechanics and Dynamical Astronomy》1976,14(1):11-17
A selective survey of then-body problem of celestial mechanics is given where the emphasis is on the asymptotic behavior of all solutions ast, the possible configurations the particles can assume in phase space and in physical space, and collision and non-collision singularities.Supported in part by NSF Grant MPS 71-03407 A03. 相似文献
15.
R. Broucke 《Celestial Mechanics and Dynamical Astronomy》1971,4(1):110-115
A Recurrent Power Series solution is given for the classicalN-body problem. The application to numerical integration is also pointed out. 相似文献
16.
How the Method of Minimization of Action Avoids Singularities 总被引：4，自引：0，他引：4
C. Marchal 《Celestial Mechanics and Dynamical Astronomy》2002,83(1-4):325-353
The method of minimization of action is a powerful technique of proving the existence of particular and interesting solutions of the n-body problem, but it suffers from the possible interference of singularities. The minimization of action is an optimization and, after a short presentation of a few optimization theories, our analysis of interference of singularities will show that:(A) An n-body solution minimizing the action between given boundary conditions has no discontinuity: all n-bodies have a continuous and bounded motion and thus all eventual singularities are collisions;(B) A beautiful extension of Lambert's theorem shows that, for these minimizing solutions, no double collision can occur at an intermediate time;(C) The proof can be extended to triple and to multiple collisions. Thus, the method of minimization of action leads to pure n-body motions without singularity at any intermediate time, even if one or several collisions are imposed at initial and/or final times.This method is suitable for non-infinitesimal masses only. Fortunately, a similar method, with the same general property with respect to the singularities, can be extended to n-body problems including infinitesimal masses. 相似文献
17.
S. Terracini 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):3-25
This expository paper gathers some of the results obtained by the author in recent works in collaboration with Davide Ferrario and Vivina Barutello, focusing on the periodic n-body problem from the perspective of the calculus of variations and minimax theory. These researches were aimed at developing a systematic variational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems and achievements of this approach. The material here was exposed in the talk that given at the Meeting CELMEC IV promoted by SIMCA (Società italiana di Meccanica Celeste). 相似文献
18.
R. H. Miller 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):19-26
Computation and a wealth of new observational techniques have reinvigorated dynamical studies of galaxies and star clusters. These objects are examples of the gravitationaln-body problem withn in the range from a few hundred to 1011. Relaxation effects dominate at the low end and are completely negligible at the high end. The gravitationaln-body problem is chaotic, and the principal challenge in doing physics where that problem is involved (whether computationally or with analytic theory) is to ensure that chaos has not vitiated the results. Enforcing a Liouville theorem accomplishes this with collision-free large-n problems, but equivalent recipes are not in common use for smallern. We describe some important insights and discoveries that have come from computation in stellar dynamics, discuss chaos briefly, and indicate the way the physics that comes up in different astronomical contexts is addressed in numerical methods currently in use. Graphics is a vital part of any computational approach. The long range prospects are very promising for continued high scientific productivity in stellar dynamics. 相似文献
19.
G. Mingotti F. Topputo F. Bernelli-Zazzera 《Celestial Mechanics and Dynamical Astronomy》2011,110(2):169-188
In this paper novel Earth–Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun–Earth
problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed
in the Sun–Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory
is optimized in the full n-body problem which takes into account planets’ orbital inclinations and eccentricities. Accurate, efficient solutions with
reasonable flight times are presented and compared with known results. 相似文献