PRECESSION AND CHAOS IN THE CLASSICAL TWO-BODY PROBLEM IN A SPHERICAL UNIVERSE

2008 ◽  
Vol 18 (02) ◽  
pp. 455-464 ◽  
Author(s):  
JOHN F. LINDNER ◽  
MARTHA I. ROSEBERRY ◽  
DANIEL E. SHAI ◽  
NICHOLAS J. HARMON ◽  
KATHERINE D. OLAKSEN
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
Flat Space ◽  
Three Body Problem ◽  
Two Body Problem ◽  
Extreme Sensitivity ◽  
Three Body ◽  
Spherical Curvature ◽  
Spherical Universe ◽  
Two Bodies

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

scholarly journals Quasi-Lntegrals of the Plane Restricted Three-Body Problem

1993 ◽  
Vol 132 ◽  
pp. 309-319
Author(s):  
E.M. Nezhinskij
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Test Body ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Jacobi Integral ◽  
Three Body ◽  
Two Bodies

AbstractThe paper is concerned with studying the domain of possible motion and a field of the test body velocities in the plane restricted problem of three bodies. The study is based on existence of a quasi-integral of areas (similar to an integral of areas in the problem of two bodies) as well as on the Jacobi integral. The method of constructing the quasi-integrals is a standard one (see, for example, [1],[2].

scholarly journals Initial Parameter Analysis about Resonant Orbits in Earth-Moon System

2019 ◽  
Vol 2019 ◽  
pp. 1-17
Author(s):  
Li-Bo Liu ◽  
Ying-Jing Qian ◽  
Xiao-Dong Yang
Keyword(s):  
Body Problem ◽  
Parameter Analysis ◽  
Initial Parameter ◽  
The Moon ◽  
Three Body Problem ◽  
Ballistic Capture ◽  
Moon System ◽  
Two Body Problem ◽  
Resonant Orbits ◽  
Three Body

The initial parameters about resonant orbits in the Earth-Moon system were investigated in this study. Resonant orbits with different ratios are obtained in the two-body problem and planar circular restricted three-body problem (i.e., PCRTBP). It is found that the eccentricity and initial phase are two important initial parameters of resonant orbits that affect the closest distance between the spacecraft and the Moon. Potential resonant transition or resonant flyby may occur depending on the possibility of the spacecraft approaching the Moon. Based on an analysis of ballistic capture and flyby, the Kepler energy and the planet’s perturbed gravitational sphere are used as criteria to establish connections between the initial parameters and the possible “steady” resonant orbits. The initial parameter intervals that can cause instability of the resonant orbits in the CRTBP are obtained. Examples of resonant orbits in 1:2 and 2:1 resonances are provided to verify the proposed criteria.

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.

scholarly journals Symbolic Dynamics, Mixing and Entropy in the Three-Body Problem

2016 ◽  
Vol 25 (3) ◽  
Author(s):  
A. Mylläri ◽  
V. Orlov ◽  
A. Chernin ◽  
A. Martynova ◽  
T. Mylläri
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Initial Conditions ◽  
Free Fall ◽  
Equal Mass ◽  
Binary Collision ◽  
Three Body Problem ◽  
Symbolic Sequences ◽  
Three Body ◽  
Sensitivity To Initial Conditions

AbstractWe use symbolic dynamics in the classic equal-mass free-fall three-body problem. Different methods for constructing symbolic sequences (in the process of numerical integration of trajectories) allow one to demonstrate (and illustrate on the Agekian-Anosova map) sensitivity to initial conditions, estimate entropies (Shannon, Markov and others), plot binary collision curves, reveal systems with intensive triple interactions (interplay), etc.

STABILIZATION OF CHAOTIC BEHAVIOR IN THE RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (10) ◽  
pp. 3603-3606 ◽  
Author(s):  
ARSEN DZHANOEV ◽  
ALEXANDER LOSKUTOV
Keyword(s):  
Body Problem ◽  
Chaotic Behavior ◽  
Melnikov Method ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lagrange Points ◽  
Three Body ◽  
Two Bodies

The restricted three-body problem on the example of a perturbed Sitnikov case is considered. On the basis of the Melnikov method we study a possibility to stabilize the obtained chaotic solutions by two bodies placed in the triangular Lagrange points. It is shown that in this case, in addition to regular and chaotic solutions, there exist stabilized solutions.

scholarly journals Libration of Retrograde Satellite Orbits in the Circular Plane Restricted Three-Body Problem

1979 ◽  
Vol 81 ◽  
pp. 41-44
Author(s):  
Daniel Benest
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
The Body ◽  
Satellite Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Limited Region ◽  
Lagrange Points ◽  
Three Body ◽  
Circular Plane

In the circular plane restricted three-body problem, we study the stable large retrograde non-periodic satellite orbits. We use rotating axes with the origin in the body around which turns the satellite, called its primary. We choose the initial conditions such as Yo=0 and Uo=0, so that an orbit can be represented by a point in the (Xo,Vo) plane. In this plane, the set of stable orbits is represented by a limited region, which we call the stability zone. This zone is composed in general by a large continental region, approximately limited by Lagrange points, and a peninsula more or less elongated.

scholarly journals Stability of the Triangular Lagrangian Solutions of the Photo Gravitational Restricted Three-Body Problem in the Three-Dimensional Case

1993 ◽  
Vol 132 ◽  
pp. 291-308
Author(s):  
Md. Ghulam Murtuza ◽  
Vijay Kumar ◽  
R.K. Choudhry
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
The Stability ◽  
Three Body

AbstractThe stability of the triangular Lagrangian solutions for the photo-gravitational restricted three-body problem in the three-dimensional case is investigated for the case when the resonances are absent and also when the resonances are present. Stability is proved for most (in the sense of Lebesgue) initial conditions for all μ < μ0 except for the resonance cases.

scholarly journals Baryon Structure and QCD: Nucleon Calculations

1989 ◽  
Vol 42 (2) ◽  
pp. 147 ◽  
Author(s):  
CJ Burden ◽  
RT Cahill ◽  
J Praschifka
Keyword(s):  
Form Factor ◽  
Body Problem ◽  
Chiral Limit ◽  
Nucleon Mass ◽  
Three Body Problem ◽  
Two Body Problem ◽  
Functional Methods ◽  
The One ◽  
Three Body ◽  
Constituent Mass

We present numerical calculations for the structure and mass of the i + nucleon in the chiral limit, using a covariant, QCD based formalism developed previously. The three-body problem of quarks interacting'via gluon exchange is treated as a quark-diquark two-body problem. The nucleon mass and a nucleon-quark-diquark form factor are determined as a function of the one parameter, the diquark form factor normalisation, which can be determined by functional methods. The constituent mass the unpaired quark within the nucleon is estimated to be about 0.44 Gey.

scholarly journals Construction of Periodic Orbits, Problems of Stability and Period Determination, in the Elliptical Non-Planar Restricted Problem

1983 ◽  
Vol 74 ◽  
pp. 257-261
Author(s):  
Colette Edelman
Keyword(s):  
Numerical Integration ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Body Problem ◽  
Initial Conditions ◽  
Monodromy Matrix ◽  
Second Phase ◽  
Three Body Problem ◽  
Fixed Frame ◽  
Three Body

AbstractPeriodic orbits in a fixed frame are constructed in the vicinity of nonperiodic solutions of the non perturbed problem. In a first phase, approximate initial conditions are found and in a second phase more accurate initial conditions obtained are used in order to check the periodic orbit by numerical integration of the three-body problem. Some peculiar solutions are found, for example, orbit with nearly zero angular momentum. A study of stability of periodic solutions is proposed with an approximation of the monodromy matrix Φ (T,o),not requiring numerical integration of the 6x6 variational linear system. Finally, some numerical problems of period determination are outlined.

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