Perihelia reduction and global Kolmogorov tori in the planetary problem

TL;DR

This paper proves the existence of a large set of quasi-periodic motions in the planetary problem with high eccentricities and inclinations, extending previous results that required smallness assumptions.

Contribution

It introduces a new coordinate system that reduces the system's degrees of freedom and reflection invariance, enabling explicit construction of a nearly integrable system without Birkhoff normal form.

Findings

Existence of quasi-periodic motions with high eccentricities and inclinations.

New coordinate system simplifies the planetary problem analysis.

Explicit construction of a near-integrable system.

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 [2, Ch III, \S 1, n. 6, p. 128] 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.