scholarly journals ON THE BIFURCATION AND CONTINUATION OF PERIODIC ORBITS IN THE THREE BODY PROBLEM

2011 ◽  
Vol 21 (08) ◽  
pp. 2211-2219 ◽  
Author(s):  
K. I. ANTONIADOU ◽  
G. VOYATZIS ◽  
T. KOTOULAS
Keyword(s):  
Periodic Orbits ◽  
General Problem ◽  
Restricted Problem ◽  
Body Problem ◽  
Three Body Problem ◽  
Numerical Studies ◽  
Circular Problem ◽  
Families Of Periodic Orbits ◽  
Elliptic Restricted Problem ◽  
Three Body

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem, asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These families of periodic orbits continue existing and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario first described by Bozis and Hadjidemetriou.

Families of periodic orbits in the planar three-body problem

1975 ◽  
Vol 12 (2) ◽  
pp. 175-187 ◽  
Author(s):  
John D. Hadjidemetriou ◽  
Th. Christides
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Three Body Problem ◽  
Families Of Periodic Orbits ◽  
Three Body

scholarly journals A Catalogue of Periodic Orbits in the Elliptic Restricted 3-Body Problem

1992 ◽  
Vol 152 ◽  
pp. 171-174 ◽  
Author(s):  
R. Dvorak ◽  
J. Kribbel
Keyword(s):  
Periodic Orbits ◽  
Restricted Problem ◽  
Body Problem ◽  
Grid Size ◽  
Mass Ratios ◽  
Families Of Periodic Orbits ◽  
Elliptic Restricted Problem ◽  
The Stability

Results of families of periodic orbits in the elliptic restricted problem are shown for some specific resonances. They are calculated for all mass ratios 0 < μ < 1.0 of the primary bodies and for all values of the eccentricity of the orbit of the primaries e < 1.0. The grid size is of 0.01 for both parameters. The classification of the stability is undertaken according to the usual one and the results are compared with the extensive studies by Contopoulos (1986) in different galactical models.

scholarly journals Integrals of Motion

1975 ◽  
Vol 69 ◽  
pp. 209-225 ◽  
Author(s):  
G. Contopoulos
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Threshold Energy ◽  
Three Body Problem ◽  
Integrals Of Motion ◽  
Surface Of Section ◽  
Families Of Periodic Orbits ◽  
Stable Periodic ◽  
Three Body

The properties of conservative dynamical systems of two or more degrees of freedom are reviewed. The transition from integrable to ergodic systems is described. Nonintegrability is due to the interaction of two, or more, resonances. Then one sees, on a surface of section, infinite types of islands of various orders, while the asymptotic curves from unstable invariant points intersect each other along homoclinic and heteroclinic points producing an apparent ‘dissolution’ of the invariant curves. A threshold energy is defined separating near integrable systems from near ergodic ones. The possibility of real ergodicity for large enough energies is discussed. In the case of many degrees of freedom we also distinguish between integrable, ergodic, and intermediate cases. Among the latter are systems of particles interacting with Lennard-Jones interparticle potential. A threshold energy was derived, which is proportional to the number of particles. Finally some recent results about the general three-body problem are described. One can extend the families of periodic orbits of the restricted problem to the general three-body problem. Many of these orbits are stable. An empirical study of orbits near the stable periodic orbits indicates the existence of 2 integrals of motion besides the energy.

scholarly journals Three-body problem - from Newton to supercomputer plus machine learning

2021 ◽  
Author(s):  
Shijun LIAO ◽  
Xiaoming Li ◽  
Yu Yang
Keyword(s):  
Machine Learning ◽  
Periodic Orbits ◽  
High Performance ◽  
Body Problem ◽  
Ann Model ◽  
Three Body Problem ◽  
Starting Point ◽  
Families Of Periodic Orbits ◽  
Artificial Neural Network Ann ◽  
Three Body

Abstract The famous three-body problem can be traced back to Newton in 1687, but quite few families of periodic orbits were found in 300 years thereafter. In this paper, we propose an effective approach and a roadmap to numerically gain planar periodic orbits of three-body systems with arbitrary masses by means of machine learning based on an artificial neural network (ANN) model. Given any a known periodic orbit as a starting point, this approach can provide more and more periodic orbits (of the same family name) with variable masses, while the mass domain having periodic orbits becomes larger and larger, and the ANN model becomes wiser and wiser. Finally, we have an ANN model trained by means of all obtained periodic orbits of the same family, which provides a convenient way to give accurate enough predictions of periodic orbits with arbitrary masses for physicists and astronomers. It suggests that the high-performance computer and artificial intelligence (including machine learning) should be the key to gain periodic orbits of the famous three-body problem.

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