Isosceles Three-body Problem Research Articles

Minimal Geodesics of the Isosceles Three Body Problem

The isosceles three-body problem with nonnegative energy is studied from a variational point of view based on the Jacobi–Maupertuis metric. The solutions are represented by geodesics in the two-dimensional configuration space. Since the metric is singular at collisions, an approach based on the theory of length spaces is used. This provides an alternative to the more familiar approach based on the principle of least action. The emphasis is on the existence and properties of minimal geodesics, that is, shortest curves connecting two points in configuration space. For any two points, even singular points, a minimal geodesic exists and is nonsingular away from the endpoints. For the zero energy case, it is possible to use knowledge of the behavior of the flow on the collision manifold to see that certain solutions must be minimal geodesics. In particular, the geodesic corresponding to the collinear homothetic solution turns out to be a minimal for certain mass ratios.

On the Manev spatial isosceles three-body problem

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all energy levels. We find that orbits tending to/ejecting from total collision are present for a large set of angular momenta. We also discover that as the angular momentum is increased, the collision manifold changes its topology.

New Phenomena in the Spatial Isosceles Three-Body Problem with Unequal Masses

This work studies periodic orbits as action minimizers in the spatial isosceles three-body problem with mass [Formula: see text]. In each period, the body with mass [Formula: see text] moves up and down on a vertical line, while the other two bodies have the same mass [Formula: see text], and rotate about this vertical line symmetrically. For given [Formula: see text], such periodic orbits form a one-parameter set with a rotation angle [Formula: see text] as the parameter. [Formula: see text]Two new phenomena are found for this set. First, for each [Formula: see text], this set of periodic orbits bifurcate from a circular Euler (central configuration) orbit to a Broucke (collision) orbit as [Formula: see text] increases from [Formula: see text] to [Formula: see text]. There exists a critical rotation angle [Formula: see text], where the orbit is a circular Euler orbit if [Formula: see text]; a spatial orbit if [Formula: see text]; and a Broucke (collision) orbit if [Formula: see text]. The exact formula of [Formula: see text] is numerically proved to be [Formula: see text]. Second, oscillating behaviors occur at rotation angle [Formula: see text] for all [Formula: see text]. Actually, the orbit with [Formula: see text] runs on its initial periodic shape for only a few periods. It breaks the first periodic shape and becomes irregular in a moment. However, it runs close to a different periodic shape after a while. In a short time, it falls apart from the second periodic shape and runs irregularly again. Such oscillation continues as time [Formula: see text] increases. Up to [Formula: see text], the orbit is bounded and keeps oscillating between periodic shapes and irregular motions. Further study implies that, for each [Formula: see text], the angle between any two consecutive periodic shapes is a constant. When [Formula: see text], similar oscillating behaviors are expected.

Periodic brake orbits in the planar isosceles three-body problem

A brake orbit is an orbit that starts with zero initial velocity. The purpose of this paper is to find periodic brake orbits in the isosceles three-body problem. We use various shooting arguments; we follow a curve of brake initial conditions under the flow until it reaches a suitable surface, and show that the image curve includes a point that corresponds to a periodic orbit. As a result, we prove the existence of six types of periodic brake orbits.

Quasiperiodic Collision Solutions in the Spatial Isosceles Three-Body Problem with Rotating Axis of Symmetry

We consider the spatial isosceles Newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) located symmetrically with respect to this plane. Using variational methods, we find a one-parameter family of collision solutions for this system. All these solutions are periodic in a rotating frame.

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

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.

Minimizing Periodic Orbits with Regularizable Collisions in the n-Body Problem

In the Newtonian n-body problem, there are various subsystems with two degrees of freedom, such as the collinear three-body problem and the isosceles three-body problem. After we determine a normal form of the Lagrangians of these subsystems, we prove the existence of periodic solutions with regularizable collisions for these systems. Our result includes several examples, such as Schubart’s orbit with or without equal masses, among others.

Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem

We study the isosceles three-body problem and show that there exist infinitely many families of relative periodic orbits converging to heteroclinic cycles between equilibria on the collision manifold in Devaney's blown-up coordinates. Towards this end, we prove that two types of heteroclinic orbits exist in much wider parameter ranges than previously detected, using self-validating interval arithmetic calculations, and we appeal to the previous results on heteroclinic orbits. Moreover, we give numerical computations for heteroclinic and relative periodic orbits to demonstrate our theoretical results. The numerical results also indicate that the two types of heteroclinic orbits and families of relative periodic orbits exist in wider parameter regions than detected in the theory and that some of them are related to Euler orbits.

The restricted planar isosceles three-body problem with non-negative energy

We consider a restricted three-body problem consisting of two positive equal masses m1 = m2 moving, under the mutual gravitational attraction, in a collision orbit and a third infinitesimal mass m3 moving in the plane P perpendicular to the line joining m1 and m2. The plane P is assumed to pass through the center of mass of m1 and m2. Since the motion of m1 and m2 is not affected by m3, from the symmetry of the configuration it is clear that m3 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 m3 when its angular momentum is different from zero and the motion of m1 and m2 is not periodic. Our main result is the characterization of the global flow of this problem.

Free-fall and heteroclinic orbits to triple collisions in the isosceles three-body problem

We consider the spatial isosceles three-body problem where two masses are equal and the other may be different. We discuss free-fall orbits in the planar case and show that there exists a countable family of orbits converging to triple collisions forward and backward in time. In Devaney's coordinates, the orbits correspond to topologically transverse heteroclinic orbits between equilibria on the collision manifold in the blown-up equations (Theorem 1.1). By the "window theory" of Moeckel [SIAM J. Math. Anal., 15 (1984), 857-876], we see that there exists an orbit shadowing a prescribed path on the graph consisting of these new heteroclinic orbits and already known, heteroclinic and periodic orbits. In particular, choosing appropriate infinite paths, we obtain a variety of new oscillatory orbits.

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

We study the Anisotropic Stormer Problem (ASP) and the Isosceles Three-Body Problem (IP), from the viewpoint of integrability, using Morales–Ramis theory and its generalization. The study of their integrability presents particular interest since they model important physical phenomena. Both problems can be reduced with respect to the S 1 symmetry. Almeida and Stuchi [M.A. Almeida, T.J. Stuchi, Non-integrability of the anisotropic Stormer problem with angular momentum, Physica D 189 (2004) 219–233] proved that the reduced ASP is non-integrable for almost all values of the parameters. In this paper we establish the non-integrability (in the extended Liouville sense) of the remaining cases. The IP is a special case of the three-body problem and it can be considered as a generalization of the Sitnikov problem. Here we prove that the complexified reduced IP does not admit an additional independent meromorphic first integral.

Characteristics of motions in the 2D isosceles three-body problem with unequal masses

We analyze the general 2D isosceles three-body problem for various ratios ɛ of the mass of the central body to the mass of each of the other two bodies. We set the initial conditions using two parameters: the virial coefficient k and the parameter \(\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }\), where \(\dot r\) is the relative velocity of the two outer bodies and \(\dot R\) is the velocity of the central body relative to the center of mass of the outer bodies. We compare statistical dependences between evolutionary parameters of triple systems with various values of ɛ, and analyze the k and μ dependences of the number of crossings of the center of mass of the triple system by the central body and the lifetime of the system. We construct the functions Rmax(rmax), where rmax and Rmax are the maximum achievable distances between the outer bodies, and between the central body and the center of mass of the outer bodies in the triple system. The parameter ɛ proves to be the most important parameter of the problem, and determines the relationship between the measures of the regular and stochastic trajectories. However, there exist “seeds” of stochasticity, even at small ɛ∼10−2. The measure of the stochastic orbits increases with ɛ; when ɛ≥10, virtually the entire region of the initial conditions corresponds to stochastic trajectories.

Classification of orbits in the plane isosceles three-body problem

The general plane isosceles three-body problem is considered for different ratios of the central body mass to the masses of other bodies. The central body goes through the middle of the segment connecting the other bodies along the perpendicular to this segment. The initial conditions are chosen by two parameters: the virial ratio k and the parameter μ =, where r is the relative velocity of the 'outer' bodies, and R is the velocity of the 'central' body with respect to the mass centre of the 'outer' bodies. The equations of motion are numerically integrated until one of three times: the time of escape of the central body, its time of ejection with R > 100d, or 1000τ (here d is the mean size, and T is the mean crossing time of the triple system). The regions corresponding to escapes of the central body after different numbers of triple approaches are found at the plane of parameters k E (0,1) and μ∈ (-1, 1). The regions of stable motions are revealed. The zones of regular and stochastic orbits are outlined. The fraction of stochastic trajectories increases with the central mass. The fraction of stable orbits is highest for equal masses of the bodies.

Heteroclinic Orbits and Bernoulli Shift¶for the Elliptic Collision Restricted¶Three-Body Problem

We consider two mass points of masses m1=m2=\(\) moving under Newton's law of gravitational attraction in a collision elliptic orbit while their centre of mass is at rest. A third mass point of mass m3≈ 0, moves on the straight line L, perpendicular to the line of motion of the first two mass points and passing through their centre of mass. Since m3≈ 0, the motion of the masses m1 and m2 is not affected by the third mass, and from the symmetry of the motion it is clear that m3 will remain on the line L. So the three masses form an isosceles triangle whose size changes with the time. The elliptic collision restricted isosceles three-body problem consists in describing the motion of m3.

A Global Analysis of The Generalized Sitnikov Problem

AbstractThe isosceles three-body problem with Sitnikov-type symmetry has been reduced to a two-dimensional area-preserving Poincaré map depending on two parameters: the mass ratio, and the total angular momentum. The entire parameter space is explored, contrasting new results with ones obtained previously in the planar (zero angular momentum) case. The region of allowable motion is divided into subregions according to a symbolic dynamics representation. This enables a geometric description of the system based on the intersection of the images of the subregions with the preimages. The paper also describes the regions of allowable motion and bounded motion, and discusses the stability of the dominant periodic orbit.

Keplerian periodic orbits in the isosceles problem

The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision. The basic tool used is the study of the Poincare maps associated with the periodic solutions.

Order and chaos in the planar isosceles three-body problem

The planar isosceles three-body problem has been reduced to a two-dimensional area preserving Poincare map f. Using certain symmetry properties of the underlying differential equations and numerical integration, we offer a global description of f in the case of three equal masses. This description, which is based on the mapping of areas, immediately leads to the existence of various types of motion such as capture-escape, permanent capture, ejection-collision, etc., and their corresponding measures in the map domain. Moreover, this technique readily allows one to distinguish between so-called "fast" and "chaotic" scattering. Although capture-escape is the subset with the highest measure, there exist two important distinct invariant subsets under f where the solutions neither are captured nor lead to escape. The first set is a Cantor set which has zero measure and it is the outcome of the fact that f acts similar to the Smale horseshoe map in part of the domain. On this subset the action of f is chaotic. The second subset is an invariant region with positive measure surrounding an elliptic fixed point. In this region f acts essentially as a perturbed twist mapping where the iterates of f for the points in a large subset move on invariant curves in an orderly manner. In an appendix we cast our results in the framework of the widely studied isosceles triple collision manifold. (c) 1998 American Institute of Physics.

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

We study the global bifurcation diagram of the two-parameter family of ODE’s that govern the charged isosceles three-body problem. (The classic isosceles three-body problem and the anisotropic Kepler problem (two bodies) are included in the same family.) There are two major sources of periodic orbits. On the one hand the “Kepler” orbit, a stable orbit exhibiting the generic bifurcations as the multiplier crosses rational values. This orbit turns out to be the continuation of the classical circular Kepler orbit. On the other extreme we have the collision-ejection orbit which exhibits an “infinite-furcation.” Up to a limiting value of the parameter we have finitely many periodic orbits (for each fixed numerator in the rotation number), passed this value there is a sudden birth of an infinite number of them. We find that these two bifurcations are remarkably connected forming the main “skeleton” of the global bifurcation diagram. We conjecture that this type of global connection must be present in related problems such as the classic isosceles three-body problem and the anisotropic Kepler problem.

Global Phase Structure of the Restricted Isosceles Three-Body Problem with Positive Energy

We study a restricted three-body problem with special symmetries: the restricted isosceles three-body problem. For positive energy the energy manifold is partially compactified by adding boundary manifolds corresponding to infinity and triple collision. We use a new set of coordinates which are a variation on the McGehee coordinates of celestial mechanics. These boundary manifolds are used to study the global phase structure of this gradational system. The orbits are classified by intersection number, that is the number of times the infinitesimal body cross the line of syzygy before escaping to infinity.

Global phase structure of the restricted isosceles three-body problem with positive energy

We study a restricted three-body problem with special symmetries: the restricted isosceles three-body problem. For positive energy the energy manifold is partially compactified by adding boundary manifolds corresponding to infinity and triple collision. We use a new set of coordinates which are a variation on the McGehee coordinates of celestial mechanics. These boundary manifolds are used to study the global phase structure of this gradational system. The orbits are classified by intersection number, that is the number of times the infinitesimal body cross the line of syzygy before escaping to infinity.