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Perihelia reduction and global Kolmogorov tori in the planetary problem

Author: Gabriella Pinzari
Publisher: Providence, RI : American Mathematical Society, 2018
Series: Memoirs of the American Mathematical Society, 1218.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"We prove the existence of an almost full measure set of (3n - 2)-dimensional quasi-periodic motions in the planetary problem with (1 + n) masses, with eccentricities arbitrarily close to the Levi-Civita 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  Read more...
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Document Type: Book
All Authors / Contributors: Gabriella Pinzari
ISBN: 9781470441029 1470441020
OCLC Number: 1089116530
Notes: "September 2018. Volume 255. Number 1218 (first of 7 numbers)."
Description: v, 92 pàgines
Contents: Conté: Background and results --
Kepler maps and the Perihelia reduction --
The P-map and the planetary problem --
Global Kolmogorov tori in the planetary problem --
Proofs
Series Title: Memoirs of the American Mathematical Society, 1218.
Responsibility: Gabriella Pinzari.

Abstract:

"We prove the existence of an almost full measure set of (3n - 2)-dimensional quasi-periodic motions in the planetary problem with (1 + n) masses, with eccentricities arbitrarily close to the Levi-Civita 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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