Unchained polygons and the N-body problem

Autor: Jacques Féjoz, Alain Chenciner
Přispěvatelé: Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Centre National de la Recherche Scientifique (CNRS), UFR Mathématiques [Sciences] - Université Paris Cité (UFR Mathématiques UPCité), Université Paris Cité (UPCité), Astronomie et systèmes dynamiques (ASD), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de Paris, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Jazyk: angličtina
Rok vydání: 2009
Předmět:
Zdroj: Regular and Chaotic Dynamics
Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2009, 14 (1), pp.64-115
Regular and Chaotic Dynamics, 2009, 14 (1), pp.64-115
Regular and Chaotic Dynamics, 2009, 14, pp.64-115. ⟨10.1134/S1560354709010079⟩
ISSN: 1560-3547
1468-4845
DOI: 10.1134/S1560354709010079⟩
Popis: International audience; We study both theoretically and numerically the Lyapunov families which bifurcate in the vertical direction from a horizontal relative equilibrium in R3. As explained in [1], very symmetric relative equilibria thus give rise to some recently studied classes of periodic solutions. We discuss the possibility of continuing these families globally as action minimizers in a rotating frame where they become periodic solutions with particular symmetries. A first step is to give estimates on intervals of the frame rotation frequency over which the relative equilibrium is the sole absolute action minimizer: this is done by generalizing to an arbitrary relative equilibrium the method used in [2] by V. Batutello and S. Terracini. In the second part, we focus on the relative equilibrium of the equal-mass regular N-gon. The proof of the local existence of the vertical Lyapunov families relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We compute the symmetry groups G r/s ( N, k, eta) of the vertical Lyapunov families observed in appropriate rotating frames, and use them for continuing the families globally. The paradigmatic examples are the ``Eight'' families for an odd number of bodies and the ``Hip- Hop'' families for an even number. The first ones generalize Marchal's P 12 family for 3 bodies, which starts with the equilateral triangle and ends with the Eight [1, 3-6]; the second ones generalize the Hip-Hop family for 4 bodies, which starts from the square and ends with the Hip-Hop [1, 7, 8]. We argue that it is precisely for these two families that global minimization may be used. In the other cases, obstructions to the method come from isomorphisms between the symmetries of different families; this is the case for the so-called ``chain'' choreographies (see [6]), where only a local minimization property is true (except for N = 3). Another interesting feature of these chains is the deciding role played by the parity, in particular through the value of the angular momentum. For the Lyapunov families bifurcating from the regular N-gon whith N
Databáze: OpenAIRE
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