The stability of periodic orbits in the three-body problem

Hadjidemetriou, John D.

doi:10.1007/bf01228563

Cited by 108 publications

(42 citation statements)

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“…This mapping is fourdimensional and the method is similar to that used in Hadjidemetriou (1975) for the general three-body problem. We have, in general, two nonunit pairs of eigenvalues, because the system is nonautonomous.…”

Section: (B) Elliptic Problemmentioning

confidence: 99%

The elliptic restricted problem at the 3 : 1 resonance

Hadjidemetriou

1992

Celestial Mech Dyn Astr

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Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, Ic~ IIc bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I~, ~, from the stable resonant 3 : i family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families L, IL, but is poor for the families L, IL. A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem.

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Section: (B) Elliptic Problemmentioning

confidence: 99%

The elliptic restricted problem at the 3 : 1 resonance

Hadjidemetriou

1992

Celestial Mech Dyn Astr

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“…An important result concerning the existence of periodic solutions in the general three-body problem was obtained by Hadjidemetriou [1975a], who proved that any symmetric periodic orbit of the CR3BP could be continued analytically to a periodic orbit in the planar general three-body problem [Hadjidemetriou, 1975b]. Using this analytical result, families of periodic orbits in the planar general three-body problem were constructed [Bozis and Hadjidemetriou, 1976] and their stability was investigated [Hadjidemetriou and Christides, 1975].…”

Section: Other Periodic Solutionsmentioning

confidence: 99%

The three-body problem

Musielak

Quarles

2014

Rep. Prog. Phys.

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Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.

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“…In the 300 years since this "three-body problem" [1] was first recognized, only three families of periodic solutions had been found, until 2013 wheň Suvakov and Dmitrašinović [4] made a breakthrough to find 13 new distinct periodic collisionless orbits belonging to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. Before their elegant work, only three families of periodic three-body orbits were found: (1) the Lagrange-Euler family discovered by Lagrange and Euler in 18th century; (2) the Broucke-Hadjidemetriou-Hénon family [5][6][7][8][9]; (3) the Figure-eight family, first discovered numerically by Moore [10] in 1993 and rediscovered by Chenciner and Montgomery [11] in 2000, and then extended to the rotating case [12][13][14][15]. In 2014, Li and Liao [16] studied the stability of the periodic orbits in [4].…”

Section: Introductionmentioning

confidence: 99%