40 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
Perturbation theory and canonical coordinates in celestial mechanics
KAM theory owes most of its success to its initial motivation: the
application to problems of celestial mechanics. The masterly application was
offered by V.I.Arnold in the 60s who worked out a theorem, that he named the
Fundamental Theorem (FT), especially designed for the planetary problem.
However, FT could be really used at that purpose only when, about 50 years
later, a set of coordinates constructively taking the invariance by rotation
and close-to-integrability into account was used. Since then, some progress has
been done in the symplectic assessment of the problem, and here we review such
results.Comment: 48 page
Quantitative KAM theory, with an application to the three-body problem
Based on quantitative ``{\sc kam} theory'', we state and prove two theorems
about the continuation of maximal and whiskered quasi--periodic motions to
slightly perturbed systems exhibiting proper degeneracy. Next, we apply such
results to prove that, in the three--body problem, there is a small set in
phase space where it is possible to detect both such families of tori. We also
estimate the density of such motions in proper ambient spaces.
Up to our knowledge, this is the first proof of co--existence of stable and
whiskered tori in a physical system.Comment: {\bf Key-words:} Properly-degenerate Hamiltonian, symplectic
coordinates, symmetry reductions. 34 pages, 2 figure