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2013
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# Jacques Féjoz

Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France,77 avenue Denfert-Rochereau, 75014 Paris, France
Université Paris-Dauphine – CEREMADE (UMR 7534),Observatoire de Paris – IMCCE (UMR 8028)

## Publications:

 Féjoz J. On Action-angle Coordinates and the Poincaré Coordinates 2013, vol. 18, no. 6, pp.  703-718 Abstract This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants "$\oint_\gamma pdq$". The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did. Keywords: Hamiltonian system, Lagrangian fibration, action-angle coordinates, Liouville–Arnold theorem, adiabatic invariants, Kepler problem, two-body problem, Poincaré coordinates, planetary problem, first integral, integrability, perturbation theory Citation: Féjoz J.,  On Action-angle Coordinates and the Poincaré Coordinates, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 703-718 DOI:10.1134/S1560354713060105
 Féjoz J. A Proof of the Invariant Torus Theorem of Kolmogorov 2012, vol. 17, no. 1, pp.  1-5 Abstract A variant of Kolmogorov’s initial proof is given, in terms of a group of symplectic transformations and of an elementary fixed point theorem. Keywords: Celestial Mechanics Citation: Féjoz J.,  A Proof of the Invariant Torus Theorem of Kolmogorov, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 1-5 DOI:10.1134/S1560354712010017
 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