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Perihelia Reduction and Global Kolmogorov Tori in the Planetary Problem


About this Title

Gabriella Pinzari

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 255, Number 1218
ISBNs: 978-1-4704-4102-9 (print); 978-1-4704-4813-4 (online)
DOI: https://doi.org/10.1090/memo/1218
Published electronically: June 25, 2018
Keywords:Canonical coordinates, Jacobi’s reduction, Deprit’s reduction, Perihelia reduction, symmetries, quasi-periodic motions, Arnold’s theorem on the stability of planetary motions.

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Table of Contents


Chapters

  • Chapter 1. Background and results
  • Chapter 2. Kepler maps and the Perihelia reduction
  • Chapter 3. The $\mathcal P$-map and the planetary problem
  • Chapter 4. Global Kolmogorov tori in the planetary problem
  • Chapter 5. Proofs
  • Appendix A. Computing the domain of holomorphy
  • Appendix B. Proof of Lemma 3.2
  • Appendix C. Checking the non-degeneracy condition
  • Appendix D. Some results from perturbation theory
  • Appendix E. More on the geometrical structure of the $\mathcal P$-coordinates, compared to Deprit’s coordinates

Abstract


We prove the existence of an almost full measure set of -dimensional quasi-periodic motions in the planetary problem with masses, with eccentricities arbitrarily close to the LeviâĂŞCivita 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 (1963) 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.

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