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1.
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
2.
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
3.
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
4.
New stacked central configurations for the planar 5-body problem 总被引：1，自引：0，他引：1
Jaume Llibre Luis Fernando Mello Ernesto Perez-Chavela 《Celestial Mechanics and Dynamical Astronomy》2011,110(1):43-52
A stacked central configuration in the n-body problem is one that has a proper subset of the n-bodies forming a central configuration. In this paper we study the case where three bodies with masses m
1, m
2, m
3 (bodies 1, 2, 3) form an equilateral central configuration, and the other two with masses m
4, m
5 are symmetric with respect to the mediatrix of the segment joining 1 and 2, and they are above the triangle generated by
{1, 2, 3}. We show the existence and non-existence of this kind of stacked central configurations for the planar 5-body problem. 相似文献
5.
The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL 3 is extended to the planar three-body problem with respect to the massm 3 of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL 3, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form 3=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom 3=0 andm 3=0.10 are discussed. 相似文献
6.
We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler
central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in
the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration,
the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and
fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to
the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the
three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with
four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass
at the midpoint of one of the parallel sides. 相似文献
7.
In this problem, one of the primaries of mass \(m^{*}_{1}\) is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ 1. The smaller primary of mass m 2 is an oblate body outside the shell. The third and the fourth bodies (of mass m 3 and m 4 respectively) are small solid spheres of density ρ 3 and ρ 4 respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m 2 is describing a circle around \(m^{*}_{1}\) . The masses m 3 and m 4 mutually attract each other, do not influence the motions of \(m^{*}_{1}\) and m 2 but are influenced by them. We also assume that masses m 3 and m 4 are moving in the plane of motion of mass m 2. In the paper, equilibrium solutions of m 3 and m 4 and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ 3=ρ 4. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m 3,m 4 as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ,μ 3,μ 4 and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system. 相似文献
8.
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
9.
Josep M. Cors Jaume Llibre Merce Ollé 《Celestial Mechanics and Dynamical Astronomy》2004,89(4):319-342
We study the planar central configurations of the 1 +n body problem where one mass is large and the other n masses are infinitesimal and equal. We find analytically all these central configurations when 2≤n≤4. Numerically, first we provide evidence that when n9 the only central configuration is the regular n-gon with the large mass in its barycenter, and second we provide also evidence of the existence of an axis of symmetry for
every central configuration.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
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
11.
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
12.
Alan Almeida santos 《Celestial Mechanics and Dynamical Astronomy》2004,90(3-4):213-238
We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of
symmetry for the restricted problem of n+1 bodies in dimension n-1. After that, we made the calculation of all c.c. for n=4.
In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math.
292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in
the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math.
81 ] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and
a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in
[Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a
regular tetrahedron parameterized by the value of central mass. 相似文献
13.
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
14.
Esther Barrabés Josep M. Cors Claudio Vidal 《Celestial Mechanics and Dynamical Astronomy》2017,129(1-2):153-176
We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (\(-1/r +e/r^{2}\)), \(e>0\), is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter \(\beta \), the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass. 相似文献
15.
We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses.
Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two
mass ratios as μ
1 = m
1/M
T and μ
2 = m
2/M
T, where m
1 and m
2 are the two unequal masses and M
T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four
types of four-body problems. 相似文献
16.
We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined. 相似文献
17.
Josep Maria Cors César Castilho Claudio Vidal 《Celestial Mechanics and Dynamical Astronomy》2009,103(2):163-177
We consider a restricted three-body problem consisting of two positive equal masses m
1 = m
2 moving, under the mutual gravitational attraction, in a collision orbit and a third infinitesimal mass m
3 moving in the plane P perpendicular to the line joining m
1 and m
2. The plane P is assumed to pass through the center of mass of m
1 and m
2. Since the motion of m
1 and m
2 is not affected by m
3, from the symmetry of the configuration it is clear that m
3 remains in the plane P and the three masses are at the vertices of an isosceles triangle for all time. The restricted planar isosceles three-body
problem describes the motion of m
3 when its angular momentum is different from zero and the motion of m
1 and m
2 is not periodic. Our main result is the characterization of the global flow of this problem. 相似文献
18.
We consider n bodies (with equal mass m) disposed at the vertices of a regular n-gon and rotating rigidly around an additional mass m
0(at its center) with a constant angular velocity (relative equilibrium). In the present paper, we prove results on the existence
and on the linear stability of equilibrium positions for a zero-mass particle submitted to the gravitational field generated
by the previous system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
Masayoshi Sekiguchi 《Celestial Mechanics and Dynamical Astronomy》2004,90(3-4):355-360
The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special
case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration)
and an additional particle of mass m0 at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits
only one mass parameter μ =m
0/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three
when μ ≥ μ
c
, and one when μ ≥ μ
c
. By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration
are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945. 相似文献
20.
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