Symmetries and noncollision closed orbits for planar N-body-type problems

Bessi, Ugo; Zelati, Vittorio Coti

doi:10.1016/0362-546x(91)90030-5

Cited by 46 publications

(37 citation statements)

References 6 publications

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“…Suitable symmetry groups of the Lagrangian action functional have been introduced and exploited in order to apply the aforementioned techniques to the class of all symmetric loops, to name a few, in the articles [2,3,4,10,11,12,15,18,19]. Surveys and further details on this approach can be found for example in [1,5,7,12,17].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Ferrario

2005

Arch. Rational Mech. Anal.

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Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in the three-dimensional space. First we give a finite complete list of symmetry groups fit to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are not already bound to collisions). Furthermore, we prove some results addressed to the question whether minimizers are planar or non-planar; as a consequence of the theory we will give general criteria for a symmetry group to yield planar or homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution); on the other hand we will provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic nonplanar orbits. These include the choreographic Marchal's P 12 family with equal masses -together with a less-symmetric choreographic family (which anyway probably coincides with the P 12 ).MSC Subj. Class: Primary 70F10; Secondary 70G75, 37C80, 70F16, 37J45.

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Section: Introductionmentioning

confidence: 99%

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Ferrario

2005

Arch. Rational Mech. Anal.

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“…The domain for f q is different from the one defined by Bessi and Coti Zelati [4]. First, we prove that the critical point q ∈ E for f q restricted on E is also a critical point for f q on E.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…The corresponding lower bound estimate in Bessi and Coti Zelati [4] is not correct since the symmetry breaks down when they move the binary collision to the origin and they work on their domain. Lemma 2.6 [6].…”

Section: (218)mentioning

confidence: 92%

A Noncollision Periodic Solution for N-Body Problems

Zhang

Zhou

2001

Journal of Mathematical Analysis and Applications

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“…The following symmetric assumption is motivated by the Keplero N-body problem and the symmetry introduced by Bessi and Coti Zelati in [1].…”

Section: Mi~(t)+v~v(t Xl(t)xn(t))=o X~(t) Enmentioning

confidence: 99%

Symmetric periodic noncollision solutions forN-body-type problems

Zhang

Zhou

1995

Acta Mathematica Sinica

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Abstract. Using the calculus of variations in the large, especially computing the category of the symmetric configuration space of symmetric N-body-type problems, we prove the existence of infinitely many symmetric noncollision periodic solutions about the symmetric and nonautonomous N-body-type problems under the assumptions that the symmetric potentials satisfy the strong force condition of Gordon. §1. IntroductionCalculus of variations in the large was used to study periodic solutions for N-body-type problems in the last few years. In this paper, we will consider a class of solutions of the following system of ordinary differential equations mi~(t)+v~,V(t, xl(t),...,xN(t))=O, x~(t) Enwhere m~ > 0 for all i, and V satisfies the following conditions: We will say that a function X(t) = (xl(t),... ,xg(t)) • C2 (R, (Rk) N) is a T-periodic noncoUision solution of (1) if X(t) is a T periodic solution of (1) and xi(t) ¢ zj(t) for all i ¢ j, and t • R.The following symmetric assumption is motivated by the Keplero N-body problem and the symmetry introduced by Bessi and Coti Zelati in [1].

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Symmetries and noncollision closed orbits for planar N-body-type problems

Cited by 46 publications

References 6 publications

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

A Noncollision Periodic Solution for N-Body Problems

Symmetric periodic noncollision solutions forN-body-type problems

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