Global Kolmogorov tori in the planetary N-body problem. Announcement of result

We improve a result in [L. Chierchia and G. Pinzari, Invent. Math. 2011] by proving the existence of a positive measure set of $(3n-2)$--dimensional quasi--periodic motions in the spacial, planetary $(1+n)$--body problem away from co--planar, circular motions. We also prove that such quasi--periodic motions reach with continuity corresponding $(2n-1)$--dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [V. I. Arnold. Russ. Math. Surv. 1963]). The main tool is a full reduction of the SO(3)--symmetry, which, in particular, retains symmetry by reflections and highlights a quasi--integrable structure, with a small remainder, independently of eccentricities and inclinations.Comment: 17 pages. Related papers: [V.I. Arnold. Russ. Math. Surv. 1963], [P. Robutel. Cel. Mech Dys Astr. 1995], [J. F\'ejoz. Erg. Th. Dyn Syst.2004], [G. Pinzari. PhD Dissertation, 2009; arXiv:1309.7028], [L. Chierchia and G. Pinzari. Invent. Math. 2011]. Acknowledgments and microscopic changes adde

A first integral to the partially averaged Newtonian potential of the three-body problem

We consider the partial average i.e., the Lagrange average with respect to {\it just one} of the two mean anomalies, of the Newtonian part of the perturbing function in the three--body problem Hamiltonian. We prove that such a partial average exhibits a non--trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint r\^ole (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an application and an open problem.Comment: misprints correcte

Perihelion librations in the secular three--body problem

A normal form theory for non--quasi--periodic systems is combined with the special properties of the partially averaged Newtonian potential pointed out in [15] to prove, in the averaged, planar three--body problem, the existence of a plenty of motions where, periodically, the perihelion of the inner body affords librations about one equilibrium position and its ellipse squeezes to a segment before reversing its direction and again decreasing its eccentricity (perihelion librations).Comment: 3 Figures, 30 page

Canonical coordinates for the planetary problem

In 1963, V. I. Arnold stated his celebrated Thorem on the Stability of Planetary Motions. The general proof of his wonderful statement (that he provided completely only for the particular case of three bodies constrained in a plane) turned out to be more difficult than expected and was next completed by J. Laskar, P. Robutel, M. Herman, J. F\'ejoz, L. Chierchia and the author. We refer the reader to the technical papers \cite{arnold63, laskarR95, rob95, maligeRL02, herman09, fej04, pinzari-th09, ChierchiaPi11b} for detailed information; to \cite{fejoz13, chierchia13, chierchiaPi14}, or the introduction of \cite{pinzari13} for reviews. The complete understanding of Arnold's Theorem relied on an analytic part and a geometric one, both highly non trivial. Of such two aspects, the analytic part was basically settled out since \cite{arnold63} (notwithstanding refinements next given in \cite{fejoz04, chierchiaPi10}). The geometrical aspects were instead mostly unexplored after his 1963's paper and have been only recently clarified \cite{pinzari-th09, chierchiaPi11b}. In fact, switching from the three--body case to the many--body one needed to develop new constructions, because of a dramatic degeneracy due to its invariance by rotations, which, if not suitably treated, prevents the application of Arnold's 1963's strategy. The purpose of this note is to provide a historical survey of this latter part. We shall describe previous classical approaches going back to Delaunay, Poincar\'e, Jacobi and point out more recent progresses, based on the papers \cite{pinzari-th09, chierchiaPi11a, chierchiaPi11b, chierchiaPi11c, pinzari13, pinzari14}. In the final part of the paper we discuss a set of coordinates recently found by the author which reduces completely its integrals, puts the unperturbed part in Keplerian form, preserves the symmetry by reflections and is regular for zero inclinations.Comment: 27 pages. Final version. Introduction slightly modified and misprints correcte

Regular Objects, Multiplicative Unitaries and Conjugation

The notion of left (resp. right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.Comment: 48 pages, Late