649 research outputs found
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
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 --dimensional
quasi--periodic motions in the spacial, planetary --body problem away
from co--planar, circular motions. We also prove that such quasi--periodic
motions reach with continuity corresponding --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
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
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