Gabriella Pinzari
via Trieste, 63, 35121, Padova
Dipartimento di Matematica Tullio LeviCivita
Publications:
Pinzari G.
NonQuasiPeriodic Normal Form Theory
2023, vol. 28, no. 6, pp. 841864
Abstract
We review a recent application of the ideas of normal form theory to systems
(Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate
variable. The main difference from the standard case consists in the nonuniqueness of the
normal form and the total absence of the small divisors problem. The exposition is quite general,
so as to allow extensions to the case of more nonperiodic coordinates, and more functional
settings. Here, for simplicity, we work in the realanalytic class.

Pinzari G.
Aspects of the Planetary Birkhoff Normal Form
2013, vol. 18, no. 6, pp. 860906
Abstract
The discovery of the Birkhoff normal form for the planetary manybody problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a direct proof (after the proof in [18]) of the celebrated Arnold’s Theorem [5] on the stability of planetary motions. In this paper, after reviewing the story of the proof of this theorem, we focus on technical aspects of this normal form. We develop an asymptotic formula for it that may turn to be useful in applications. Then we provide two simple applications to the threebody problem: we prove that the “density” of the Kolmogorov set of the spatial threebody problem does not depend on eccentricities and the mutual inclination but depends only on the planets’ masses and the separation among semiaxes (going in the direction of an assertion by V.I. Arnold [5]) and, using Nehorošhev Theory [33], we prove, in the planar case, stability of all planetary actions over exponentiallylong times, provided meanmotion resonances are excluded. We also briefly discuss difficulties for full generalization of the results in the paper.
