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### Open Access Journals

ERA-MS

We improve a result in [9] by proving the existence of a positive measure set of $(3n-2)$-dimensional quasi-periodic motions in the spacial, planetary $(1+n)$-body problem away from co-planar, circular motions. We also prove that such quasi-periodic motions reach with continuity corresponding $(2n-1)$-dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [2]).
The main tool is a full reduction of the SO(3)-symmetry, which retains symmetry by reflections and highlights a quasi-integrable structure, with a small remainder, independently of eccentricities and inclinations.

JMD

Birkhoff normal forms for the (secular) planetary problem are
investigated. Existence and uniqueness is discussed and it is shown that
the classical Poincaré variables and the
ʀᴘs-variables (introduced in [6]), after a
trivial lift, lead to the same Birkhoff normal form; as a corollary
the Birkhoff normal form (in Poincaré variables) is degenerate at
all orders (answering a question of M. Herman). Non-degenerate
Birkhoff normal forms for partially and totally reduced cases are
provided and an application to long-time stability of secular
action
variables (eccentricities and inclinations) is discussed.

DCDS-S

Arnold's "Fundamental Theorem'' on properly-degenerate systems [3, Chapter IV] is revisited and improved with particular attention to the relation between the perturbative parameters and to the measure of the Kolmogorov set. Relations with the planetary many-body problem are shortly discussed.

keywords:
many–body problem
,
small divisors
,
invariant tori
,
degeneracies.
,
KAM theory
,
Kolmogorov set

## Year of publication

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