Aspects of the Planetary Birkhoff Normal Form

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.