scholarly journals New Phenomena in the Spatial Isosceles Three-Body Problem

2015 ◽  
Vol 25 (09) ◽  
pp. 1550116 ◽  
Author(s):  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Initial Condition ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Straight Line ◽  
Three Body ◽  
New Phenomena ◽  
Two Bodies

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

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

2015 ◽  
Vol 25 (12) ◽  
pp. 1550169 ◽  
Author(s):  
Duokui Yan ◽  
Rongchang Liu ◽  
Xingwei Hu ◽  
Weize Mao ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Mass Formula ◽  
Body Problem ◽  
Rotation Angle ◽  
The Body ◽  
Vertical Line ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Three Body ◽  
New Phenomena

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.

scholarly journals A Complete Family of Periodic Solutions of the Planar Three-Body Problem and their Stability

1977 ◽  
Vol 33 ◽  
pp. 159-159
Author(s):  
M. Hénon
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Three Body Problem ◽  
Complete Family ◽  
First Order ◽  
Hierarchical Arrangement ◽  
The Family ◽  
Three Body

AbstractWe give a complete description of a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with a rectilinear orbit, computed by Schubart in 1956. It ends in retrograde revolution, i.e., a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the “interplay” type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.

On the generalized restricted problem of three bodies

1974 ◽  
Vol 22 ◽  
pp. 85
Author(s):  
G. N. Duboshin
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Active Material ◽  
The Body ◽  
The Other ◽  
Three Body Problem ◽  
Straight Line ◽  
Second Derivatives ◽  
Material Points ◽  
Three Body

AbstractThe particular case of the complete generalized three-body problem where one of the body-points does not exert influence on the other two is analysed. These active material points act on the passive point and also on each other with forces (attraction or repulsion), proportional to the product of masses of both points and a certain function of the time, their mutual distances and their first and second derivatives. Furthermore it is not supposed that generally the th ird axiom of mechanics (action = reaction) takes place.One determines the conditions for some particular solutions which exist, when the three points form the equilateral triangle or remain always on a straight line.Finally, some conclusions on the Liapunov stability in the simplest cases are drawn.

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 865-884 ◽  
Author(s):  
PAU ATELA ◽  
ROBERT I. McLACHLAN
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Body Problem ◽  
Kepler Problem ◽  
Global Bifurcation ◽  
Three Body Problem ◽  
Anisotropic Kepler Problem ◽  
The One ◽  
Three Body ◽  
Limiting Value

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.

scholarly journals LOCATION AND STABILITY OF THE TRIANGULAR POINTS IN THE TRIAXIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM

2021 ◽  
Vol 57 (2) ◽  
pp. 311-319
Author(s):  
M. Radwan ◽  
Nihad S. Abd El Motelp
Keyword(s):  
Linear Stability ◽  
Periodic Orbits ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Stability Regions ◽  
Triangular Points ◽  
Problem Frame ◽  
The Stability ◽  
Three Body

The main goal of the present paper is to evaluate the perturbed locations and investigate the linear stability of the triangular points. We studied the problem in the elliptic restricted three body problem frame of work. The problem is generalized in the sense that the two primaries are considered as triaxial bodies. It was found that the locations of these points are affected by the triaxiality coefficients of the primaries and the eccentricity of orbits. Also, the stability regions depend on the involved perturbations. We also studied the periodic orbits in the vicinity of the triangular points.

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