共查询到20条相似文献，搜索用时 726 毫秒
1.
This paper gives an analytic proof of the existence of Schubart-like orbit, a periodic orbit with singularities in the symmetric
collinear four-body problem. In each period of the Schubart-like orbit, there is a binary collision (BC) between the inner
two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized
and the existence is proved by using a “turning point” technique and a continuity argument on differential equations of the
regularized Hamiltonian. 相似文献
2.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons Gareth E. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):147-164
We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous
binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar
four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability
to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results
of Sweatman for linear stability with respect to collinear and symmetric perturbations. 相似文献
3.
J?rg Waldvogel 《Celestial Mechanics and Dynamical Astronomy》2012,113(1):113-123
We consider the planar symmetric four-body problem with two equal masses m 1?=?m 3?>?0 at positions (±x 1(t),?0) and two equal masses m 2?=?m 4?>?0 at positions (0, ±x 2(t)) at all times t, referred to as the rhomboidal symmetric four-body problem. Owing to the simplicity of the equations of motion this problem is well suited to study regularization of the binary collisions, periodic solutions, chaotic motion, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role. 相似文献
4.
P. Delibaltas 《Celestial Mechanics and Dynamical Astronomy》1983,29(2):191-204
In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits. 相似文献
5.
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches.
We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides
the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative
periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with
binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant
families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As
the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between
triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge
to “previously unknown” periodic orbits in the planar problem. 相似文献
6.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
7.
We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm
3 being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 相似文献
8.
K. E. Papadakis 《Astrophysics and Space Science》2009,323(3):261-272
In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m
0=β
m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov
periodic orbits, which emanate from the collinear equilibrium points L
1 and L
2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic
orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal
points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points.
All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating
homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. 相似文献
9.
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
10.
Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating
Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also
continue to this restricted problem the so called “comet orbits”.
An erratum to this article can be found at 相似文献
11.
Michael Nauenberg 《Celestial Mechanics and Dynamical Astronomy》2007,97(1):1-15
Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects
the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates
with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped
orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron.
78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family
of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight
orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for
n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the
figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given
numerically, and some examples are given of nearby stable orbits which bifurcate from these families. 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.
D. J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》1998,70(2):75-98
Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem
is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem
and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem
is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described
by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion
of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon
system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries
which orbit a larger mass.
Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange
points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases
from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied
in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the
out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped
in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun
system.
This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations.
The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation
of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario
to prove their existence is described.
This revised version was published online in July 2006 with corrections to the Cover Date. 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.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):191-207
We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally,
periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there
are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart
region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of
the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré
section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit. 相似文献
17.
K.E. Papadakis 《Astrophysics and Space Science》2004,293(3):271-287
The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary
body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain
a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators
perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient
numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic
orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the
families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
We locate members of a one-parameter family of equal-mass four-body periodic orbits in the plane. The family begins and ends with the rectilinear four-body equal-mass Schubart interplay orbit and passes through a double choreography orbit. The first-order stability of these orbits is computed. Some members of this symmetric family are stable to symmetric perturbations; however, they are unstable when all perturbations are allowed. 相似文献
19.
Abimael Bengochea Manuel Falconi Ernesto Pérez-Chavela 《Astrophysics and Space Science》2011,333(2):399-408
We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses.
The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal
to 35×10−5, corresponding to 105 times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger
influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those
horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies
follow the same path is discussed. 相似文献
20.
Joachim Schubart 《Celestial Mechanics and Dynamical Astronomy》2017,128(2-3):295-301
The paper refers to fictitious resonant orbits of planet type that surround both components of a binary system. In case of 16 studied examples a suitable choice of the starting values leads to a process of libration of special angular arguments and to an evolution with an at least temporary stay of the planet in the resonant orbit. The ratio of the periods of revolution of the binary and a planet is equal to 1:5. Eight orbits depend on the ratio 1:5 of the masses of the binary components, but two other ratios appear as well. The basis of this study is the planar, elliptic or circular restricted problem of three bodies, but remarks at the end of the text refer to a four-body problem. 相似文献