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2013
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# Alain Chenciner

Jussieu 75251 Paris Cedex 05, 75251, Paris, France
Astronomie et Systèmes Dynamiques, IMCCE, Observatoire de Paris & Départment de Mathématiques

## Publications:

 Chenciner A., Leclerc B. Between Two Moments 2014, vol. 19, no. 3, pp.  289-295 Abstract In this short note, we draw attention to a relation between two Horn polytopes which is proved in [3] as a result, on the one hand, of a deep combinatorial result in [5] and, on the other hand, of a simple computation involving complex structures. This suggests an inequality between Littlewood–Richardson coefficients, which we prove using the symmetric characterization of these coefficients given in [1]. Keywords: Littlewood–Richardson coefficients, Horn polytopes, moment maps Citation: Chenciner A., Leclerc B.,  Between Two Moments, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 289-295 DOI:10.1134/S1560354714030022
 Chenciner A., Féjoz J. Unchained Polygons and the $N$-body Problem 2009, vol. 14, no. 1, pp.  64-115 Abstract We study both theoretically and numerically the Lyapunov families which bifurcate in the vertical direction from a horizontal relative equilibrium in $\mathbb{R}^3$. 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 $G_{\frac{r}{s}}(N, k, \eta)$ 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 $N \leqslant 6$ we check in an appendix that locally the torsion is not zero, which justifies taking the rotation of the frame as a parameter. Keywords: $n$-body problem, relative equilibrium, Lyapunov family, symmetry, action minimization, periodic and quasiperiodic solutions Citation: Chenciner A., Féjoz J.,  Unchained Polygons and the $N$-body Problem, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 64-115 DOI:10.1134/S1560354709010079
 Chenciner A. A note by Poincaré 2005, vol. 10, no. 2, pp.  119-128 Abstract On November 30th 1896, Poincaré published a note entitled "On the periodic solutions and the least action principle" in the "Comptes rendus de l'Académie des Sciences". He proposed to find periodic solutions of the planar Three-Body Problem by minimizing the Lagrangian action among loops in the configuration space which satisfy given constraints (the constraints amount to fixing their homology class). For the Newtonian potential, proportional to the inverse of the distance, the "collision problem" prevented him from realizing his program; hence he replaced it by a "strong force potential" proportional to the inverse of the squared distance. In the lecture, the nature of the difficulties met by Poincaré is explained and it is shown how, one century later, these have been partially resolved for the Newtonian potential, leading to the discovery of new remarkable families of periodic solutions of the planar or spatial $n$-body problem. Keywords: Poincaré, three-body problem, action minimizing periodic solutions Citation: Chenciner A.,  A note by Poincaré , Regular and Chaotic Dynamics, 2005, vol. 10, no. 2, pp. 119-128 DOI:10.1070/RD2005v010n02ABEH000306
 Chenciner A. Collisions totales, Mouvements Complètement Paraboliques et Réduction des Homothéties Dans le Problème des $n$ corps 1998, vol. 3, no. 3, pp.  93-106 Abstract Nous étudions les propriétés du problème des n corps qui proviennent de l'homogénéité du potentiel et retrouvons dans un cadre conceptuel commun divers résultats de Sundman, McGehee et Saari. Les résultats ne sont pas nouveaux mais il nous a semblé que cette présentation les éclaire agréablement. Nous considérons des potentiels de type newtonien, homogènes de degre $2\kappa$ en la configuration. Pour n'être pas obligés de distinguer divers cas dans les inégalités, nous supposerons, ce qui inclut le cas newtonien, que $-1<\kappa<0$. Citation: Chenciner A.,  Collisions totales, Mouvements Complètement Paraboliques et Réduction des Homothéties Dans le Problème des $n$ corps, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 93-106 DOI:10.1070/RD1998v003n03ABEH000083