Gabriella Pinzari

via Trieste, 63, 35121, Padova
Dipartimento di Matematica Tullio Levi-Civita


Pinzari G.
Non-Quasi-Periodic Normal Form Theory
2023, vol. 28, no. 6, pp.  841-864
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 non-uniqueness 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 non-periodic coordinates, and more functional settings. Here, for simplicity, we work in the real-analytic class.
Keywords: normal form theory, perturbation theory, KAM theory
Citation: Pinzari G.,  Non-Quasi-Periodic Normal Form Theory, Regular and Chaotic Dynamics, 2023, vol. 28, no. 6, pp. 841-864
Pinzari G.
Aspects of the Planetary Birkhoff Normal Form
2013, vol. 18, no. 6, pp.  860-906
The discovery 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 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 three-body problem: we prove that the “density” of the Kolmogorov set of the spatial three-body problem does not depend on eccentricities and the mutual inclination but depends only on the planets’ masses and the separation among semi-axes (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 exponentially-long times, provided meanmotion resonances are excluded. We also briefly discuss difficulties for full generalization of the results in the paper.
Keywords: averaging theory, Birkhoff normal form, Nehoroshev theory, planetary many-body problem, Arnold’s Theorem on the stability of planetary motions, properly-degenerate KAM theory, steepness
Citation: Pinzari G.,  Aspects of the Planetary Birkhoff Normal Form, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 860-906

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