The first half of the present memoir is devoted to KAM theory and \textsc{Arnold}'s theorem for several planets in space, as detailed in one preprint and one article:\footnote{\url{http://people.math.jussieu.fr/~fejoz/articles.html}} -- ''Twisted conjugacies and invariant tori theorems~\cite{Fejoz:2010a}. I reprove a normal form theorem due to \textsc{Moser}~\cite{Moser:1967}, for perturbations of a vector field having a Diophantine quasi-periodic invariant torus. This normal form, which I call a \emph{twisted conjugacy}, is a gateway to invariant tori theorems of \textsc{Kolmogorov}, \textsc{Arnold}, \textsc{Rüssmann} and \textsc{Herman}, as well as to some other theorems, for example for dissipative vector fields. I introduce a \emph{hypothetical conjugacy}, i.e. a conjugacy depending on arithmetical properties of the perturbed frequency vector, as an intermediate step towards invariant tori theorems with weak non-degeneracy conditions, I improve some estimates on the functional dependance of the normal form, and give some new applications to celestial mechanics. -- ''Démonstration du théorème d'Arnold sur la stabilité du système planétaire (d'après Herman)''~\cite{Fejoz:2004}. Arnold's theorem is proved for $N$ planets in space. (The proof included in~\cite{Fejoz:2010a} is a clarification and an improvement of the abstract part of~\cite{Fejoz:2004}.) \textsc{Arnold} remarkably asserted, in the Newtonian model of the planetary problem with $N$ planets, the existence of an invariant set of positive Lebesgue measure, foliated in quasi-periodic invariant tori of dimension $3N-1$~\cite{Arnold:1963}. \textsc{Arnold}'s suggestion for proving the result in full generality was to fix the direction of the angular momentum vector, in order to get rid of a degeneracy due to the rotational invariance of the problem, and then to apply his degenerate version of Kolmogorov's theorem to find Lagrangian tori in the neighborhood of the elliptic secular singularity (circular horizontal Keplerian ellipses). This strategy of partial reduction fails because of a mysterious resonance, discovered by \textsc{Herman}, which generalizes the resonance found by \textsc{Clairaut} in the first order lunar problem. This resonance had not been noticed in the context of KAM theory because in the $2$-planet problem, \textsc{Jacobi}'s reduction of the node makes it possible to carry out the full symplectic reduction by rotations in \textsc{Delaunay}'s coordinates (I recall the definition of these coordinates in the appendix of this memoir, and give a new proof of their symplecticity). Here it is proved by induction on the number of planets, following Herman, that the local image of the frequency map of the planetary system (as a function of the semi major axes), is contained in a vector plane of codimension two, and in no vector plane of larger codimension. Using an argument of Lagrangian intersection theory, this allows us to apply an invariant tori theorem with a weak (or Rüssmann-) non-degeneracy condition. The second half of the memoir deals with periodic and relatively periodic orbits in the global many-body problem. It is based on two publications. -- ''The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium'' (with A. Chenciner)~\cite{Chenciner:2008}. It is shown that exactly two families of relatively periodic orbits bifurcate from the Lagrange equilateral triangle, namely the homographic and the $\mathcal{P}{12}$ families. Moreover, in restriction to the $4$-dimensional center manifold, the local dynamics is proved to be a twist map of an annulus of section, bounded by the two families. Another paper shows that the $\mathcal{P}{12}$ family ends at the Eight~\cite{Chenciner:2005a}. In between, the $\mathcal{P}{12}$ family is known to exist as a family of minimizers of the Lagrangian action within its symmetry class for all values of the rotation of the frame. Such a family could be non-unique, or not continuous, but numerical experiments indicate that it is not the case (see the pictures above). -- ''Unchained polygons and the {$N$}-body problem'' (with A. Chenciner)~\cite{Chenciner:2009}. The Lagrange relative equilibrium appears above as the organizing center of the Eight. We show that the same phenomenon occurs with the equal-mass relative equilibrium of the square, which appears as the organizing center of the Hip-Hop. More generally, many recently studied classes of periodic solutions bifurcate from symmetric relative equilibria. In a rotating frame where they become periodic, these families acquire remarkable symmetries. We study the possibility of continuing these families globally as action minimizers in a rotating frame, among loops sharing the same symmetries. In a preliminary step we estimate the intervals of the frame rotation frequency over which the relative equilibrium is the sole absolute action minimizer. Then we focus on our main example, 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 $G{\frac rs}(N,k,\eta)$ of the vertical Lyapunov families observed in appropriate rotating frames, and use them for continuing the families globally. Paradigmatic families are the Eight families for an odd number of bodies and the Hip-Hop families for an even number. It is precisely for these two kinds of 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, 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 with $N\leq 6$, we check in an appendix that locally the torsion is not zero, which justifies taking the rotation of the frame as a parameter. This article illustrates how fertile symmetry considerations are, not only as a proof technique, but also in the heuristic search for remarkable solutions. The many-body problem has been at the origin of numerous mathematical theories. And because of the variety of techniques its study demands, it retains all of its fascination.
