Aspects of the planetary Birkhoff normal form

Abstract

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.

44 pages. Keywords: Averaging Theory, Birkhoff normal form, Nehoro{š}ev Theory, Planetary many--body problem, Arnold's Theorem on the stability of planetary motions, Properly--degenerate kam Theory, steepness. Revised version, including Reviewer's comments. Typos corrected

References in corpus (3)

• Kolmogorov and Nekhoroshev theory for the problem of three bodies
• On the Kolmogorov set for Many-Body Problems
• Analytic Lagrangian tori for the planetary many-body problem

Cited by in corpus (5)

• Perihelion librations in the secular three--body problem
• Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations
• New Methods of Isochrone Mechanics
• Families of invariant tori in KAM theory: interplay of integer characteristics
• Perturbation theory and canonical coordinates in celestial mechanics