Families of periodic orbits in the planar three-body problem

Hadjidemetriou, John D.; Christides, Th.

doi:10.1007/bf01230210

Cited by 48 publications

(43 citation statements)

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“…Since then periodic orbits of the restricted and full three-body problem have been the subject of various numerical investigations, see e.g. [7,8,9,18,19,25] and references therein. Chenciner and Montgomery [6] proved the existence of a new type of periodic orbit of the three-body problem, namely the Figure Eight choreography.…”

Section: Introductionmentioning

confidence: 99%

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity.But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step towards a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous Figure Eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating Eight family and compute the rotating choreographies bifurcating from it.

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Section: Introductionmentioning

confidence: 99%

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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“…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]. Moreover, Katopodis [1979] demonstrated that it was possible to generalize Hadjidemetriou's result from the 3D CR3BP to the 3D general three-body problem.…”

Section: Other Periodic Solutionsmentioning

confidence: 92%

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%