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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. Are Nonsymmetric Balanced Configurations of Four Equal Masses Virtual or Real? 2017, vol. 22, no. 6, pp.  677–687 Abstract Balanced configurations of $N$ point masses are the configurations which, in a Euclidean space of high enough dimension, i.e., up to $2(N - 1)$, admit a relative equilibrium motion under the Newtonian (or similar) attraction. Central configurations are balanced and it has been proved by Alain Albouy that central configurations of four equal masses necessarily possess a symmetry axis, from which followed a proof that the number of such configurations up to similarity is finite and explicitly describable. It is known that balanced configurations of three equal masses are exactly the isosceles triangles, but it is not known whether balanced configurations of four equal masses must have some symmetry. As balanced configurations come in families, it makes sense to look for possible branches of nonsymmetric balanced configurations bifurcating from the subset of symmetric ones. In the simpler case of a logarithmic potential, the subset of symmetric balanced configurations of four equal masses is easy to describe as well as the bifurcation locus, but there is a grain of salt: expressed in terms of the squared mutual distances, this locus lies almost completely outside the set of true configurations (i. e., generalizations of triangular inequalities are not satisfied) and hence could lead most of the time only to the bifurcation of a branch of virtual nonsymmetric balanced configurations. Nevertheless, a tiny piece of the bifurcation locus lies within the subset of real balanced configurations symmetric with respect to a line and hence has a chance to lead to the bifurcation of real nonsymmetric balanced configurations. This raises the question of the title, a question which, thanks to the explicit description given here, should be solvable by computer experts even in the Newtonian case. Another interesting question is about the possibility for a bifurcating branch of virtual nonsymmetric balanced configurations to come back to the domain of true configurations. Keywords: balanced configuration, symmetry Citation: Chenciner A.,  Are Nonsymmetric Balanced Configurations of Four Equal Masses Virtual or Real?, Regular and Chaotic Dynamics, 2017, vol. 22, no. 6, pp. 677–687 DOI:10.1134/S1560354717060065
 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