arXiv subjects

Gabriella Pinzari

Publications and source records attributed to Gabriella Pinzari.

26 records · Page 2Linked to original sources

arXiv2018

Exponential stability of Euler integral in the three--body problem

The first integral characteristic of the two--centres problem is proven to be an approximate integral (in the sense of N.N.Nekhorossev) to the three--body problem, at least if the masses are very different and the particles are constrained on the same plane. The proof uses a new normal form result, carefully designed around the degeneracies of the problem, and a new study of the phase portrait of the unperturbed problem. Applications to the prediction of collisions between the two minor bodies are shown.

math-ph↗

arXiv2017

A Normal Form without small divisors

Following the techniques of [4], we formulate a Normal Form Lemma suited to close to be integrable Hamiltonian systems where not all the coordinates are action angles. The Lemma turns to be useful in the theory of KAM tori of Sun-Earth-Asteroids systems.

math.DS↗

arXiv2017

An analysis of the Sun-Earth-Asteroid systems based on the two-centre problem

We propose a new analysis of the two--centre problem particularly suited to be used as a basis to study the dynamics of Sun--Earth--Asteroid systems. Our method, based on a tricky choice of initial coordinates, allows us to evaluate the risk of collisions between the Asteroid and Earth. Moreover, it seems to be fitted to prove the existence of chains of transition tori in the planar Sun--Earth--Asteroid systems.

math.DS↗

arXiv2016

Perihelia reduction and global Kolmogorov tori in the planetary problem

We prove the existence of an almost full measure set of $(3n-2)$--dimensional quasi periodic motions in the planetary problem with $(1+n)$ masses, with eccentricities arbitrarily close to the Levi-Civita limiting value and relatively high inclinations. This extends previous results, where smallness of eccentricities and inclinations was assumed. The question had been previously considered by V.I.Arnold [2, Ch III, §1, n. 6, p. 128] in the 60s, for the particular case of the planar three--body problem, where, due to the limited number of degrees of freedom, it was enough to use the invariance of the system by the SO(3) group. The proof exploits nice parity properties of a new set of coordinates for the planetary problem, which reduces completely the number of degrees of freedom for the system (in particular, its degeneracy due to rotations) and, moreover, is well fitted to its reflection invariance. It allows the explicit construction of an associated close to be integrable system, replacing Birkhoff normal form, common tool of previous literature.

math.DS↗

arXiv2014

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. Féjoz, 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, Poincaré, 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.

math.DS↗

arXiv2014

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.

math.DS↗

arXiv2013

Aspects of the planetary Birkhoff normal form

The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{š}ev Theory [Nehoro{š}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.

math.DS↗

arXiv2013

On the Kolmogorov set for Many-Body Problems

I defended my PhD Thesis in Rome, Università Roma Tre, on April, 23, 2009, under the direction of Professor Luigi Chierchia. The judging committee was composed by Professors M. Berti, A. Celletti, C. Falcolini, J. Féjoz. Professors M. Berti and J. Féjoz refereed my thesis. The main result of my thesis is the first direct proof (the first general proof was given in [J. Féjoz, ETDS, 2004]) of a famous statement by V. I. Arnold (1963), usually referred to as "Arnold's Planetary Theorem". My proof of Arnold's Planetary Theorem relies on the rediscovery, during the year 2008, of a symplectic set of action-angle variables (described in §4 of my thesis) which perform explicitly the reduction of rotation invariance of the system. Indeed, even though in a different form, they had been previously considered by [F. Boigey, Cel. Mech. Dyn. Astr., 1982] and [A. Deprit, Cel. Mech. Dyn. Astr., 1983]. The version I found in 2008 corresponds to the "planetary" form of Boigey-Deprit variables, since it includes the elliptic elements of the instantaneous ellipses of the planets around the sun and for this reason is especially fitted to this problem . I then regularized "my" planetary variables to include co-planar and co-circular motions. This regularization leads to a set of mixed action-angle and rectangular variables analogous to Poincaré' variables but better fitted to rotation invariance of the system, since they exhibit a cyclic couple of conjugated variables. I finally applied my regularized variables to the problem, checked non-trivial torsion and obtained the proof of the theorem. I wish to thank J. Féjoz for mentioning my contribution to the proof of Arnold's Theorem, and especially my rediscovery of Deprit's reduction, in his paper [J. Fejoz, DCDS-A, 2013].

math.DS↗