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1.
Jaume Llibre 《Celestial Mechanics and Dynamical Astronomy》1990,50(1):89-96
Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m
1,..., m
n we denote by N(m
1, ..., m
n, k) the number of central configurations' of the n-body problem in k modulus dilatations and rotations. If m
n
1,..., m
n, k) is finite, then we give a bound of N(m
1,..., m
n, k) which only depends of n and k. 相似文献
2.
Rogerio Deienno Diogo Merguizo Sanchez Antonio Fernando Bertachini de Almeida Prado Georgi Smirnov 《Celestial Mechanics and Dynamical Astronomy》2016,124(4):433-456
We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric. 相似文献
3.
In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m1 lie at the vertices of a regular n-gon, n particles of mass m2 lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m0 lies at the center of mass of the system. This system admits two mass parameters μ = m0/m1 and ε = m2/m1. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε. 相似文献
4.
R. F. Arenstorf 《Celestial Mechanics and Dynamical Astronomy》1982,28(1-2):9-15
The number of equivalence classes of central configurations (abbr. c.c.) in the planar 4-body problem with three arbitrary and a fourth small mass is investigated. These c.c. are derived according to their generic origin in the 3-body problem. It is shown that each 3-body collinear c.c. generates exactly 2 non-collinear c.c. (besides 4 collinear ones) of 4 bodies with smallm
40; and that any 3-body equilateral triangle c.c. generates exactly 8 or 9 or 10 (depending onm
1,m
2,m
3) planar 4-body c.c. withm
4=0. Further, every one of these c.c. can be continued uniquely to sufficiently smallm
4>0 except when there are just 9; then exactly one of them is degenerate, and we conjecture that it is not continuable tom
4>0.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics. 相似文献
5.
Martha Alvarez-Ramírez Joaquín Delgado 《Celestial Mechanics and Dynamical Astronomy》2003,87(4):371-381
We consider the symmetric planar (3 + 1)-body problem with finite masses m
1 = m
2 = 1, m
3 = µ and one small mass m
4 = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m
4 > 0. 相似文献
6.
The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m1 and m2 in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm
1,m
2 and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System. 相似文献
7.
B. Elmabsout 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):131-151
Résumé On sait que les positions d'équilibre relatif dans le problème des trois corps, où les corps se trouvent aux sommets d'un triangle équilatéral, existent lorsque les masses sont quelconques; tandis que pourn=4 (voir [3]) et pourn=5 (voir [4]), les positions d'équilibre relatifs, où les corps se trouvent aux sommets d'un polygone régulier de n cotés, existent seulement si les masses sont égales. L'objet de cet article est de montrer que ce dernier résultat est vrai pour toutn4.
It is known that in the three body problem, the equilateral configuration of relative equilibrium exists for all values of masses, while in then-body problem, forn=4 (see [3]) andn=5 (see [4]), the position of relative equilibrium where the bodies are at the vertices of a regular polygon withn sides, exists only if the masses are equal.We prove in this paper, that this last result is true for alln4.相似文献
8.
In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented.
Three bodies of masses m
1, m
2 and m
3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the
system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction
of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families
and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal
and finally when we have three unequal masses. Series, with respect to the mass m
3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of
the mass parameters are also calculated. 相似文献
9.
In this problem of the restricted (2 + 2) bodies we have considered two magnetic dipoles of masses M
1 and M
2(M
1 > M
2) moving in circular Keplarian orbit about their centre of mass. Two minor bodies of masses m
1, m
2(m
j< M
2) are taken as electric dipoles in the field of rotating magnetic dipoles. These minor bodies interact with each other but do not perturb the primaries.We have found equations of motions which differ from that of Goudas and Petsagouraki's (1985). 相似文献
10.
Manuele Santoprete 《Celestial Mechanics and Dynamical Astronomy》2006,94(1):17-35
In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First,
in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a>2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three
bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we
recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization
of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on
the size of the configuration. 相似文献
11.
Gregory Buck 《Celestial Mechanics and Dynamical Astronomy》1991,51(4):305-317
Moulton's Theorem says that given an ordering of masses m 1, ..., m nthere exists a unique collinear central configuration. The theorem allows us to ask the questions: What is the distribution of n equal masses in the collinear central configuration? What is the behavior of the distribution as n → ∞? These questions are due to R. Moekel (personal conversation). Central configurations are found to be attracting fixed points of a flow — a flow we might call an auxiliary flow (in the text it is denoted F(X)), since it has little to do with the equations of motion. This flow is studied in an effort to characterize the mass distribution. Specifically, for a collinear central configuration of n equal masses, a bound is found for the position of the masses furthest from the center of mass. Also some facts concerning the distribution of the inner masses are discovered. 相似文献
12.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. 相似文献
13.
In this paper we prove, for all p ≥ 2, the existence of central configurations of the pn-body problem where the masses are located at the vertices of p nested regular polyhedra having the same number of vertices n and a common center. In such configurations all the masses on the same polyhedron are equal, but masses on different polyhedra
could be different. 相似文献
14.
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
15.
We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m
1 = m
2 = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m
3 < 10−3, placed initially on the z-axis. We begin by finding for the restricted problem (with m
3 = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m
3 increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of
saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m
3 ≈ 10−6, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m
3 = 0. Furthermore, as m
3 increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m
3 values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries,
as observed in the m
3 = 0 case. 相似文献
16.
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
17.
Marcelo Marchesin Deborah da Paixão Vasconcellos 《Astrophysics and Space Science》2014,352(2):443-460
We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m 5=m 6=m 7=m 8=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ 0,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability. 相似文献
18.
We consider the problem: given a collinear configuration of n bodies, find the masses which make it central. We prove that for n ≤ 6, each configuration determines a one-parameter family
of masses (after normalization of the total mass). The parameter is the center of mass when n is even and the square of the angular velocity of the corresponding circular periodic orbit when n is odd. The result is expected to be true for any n.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center
of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of
four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible
to choose three positive masses such that the given three positions with the masses form a central configuration. However
for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration.
In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region
in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of
the compact region, it is always possible to choose positive masses to make the configuration central. 相似文献
20.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1977,16(1):61-76
It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP
1 andP
2. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP
3, ...,P
k. The analytic continuation, for sufficiently small values of theN–2 bodiesP
3
...P
k
and finite values for the masses ofP
1 andP
2 has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned. 相似文献