Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result
Gabriella Pinzari
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.
keywords: Quasi-integrable structures for perturbed super-integrable systems Deprit's reduction perihelia reduction Symmetries $N$-body problem Multi-scale KAM Theory. canonical coordinates Arnold's Theorem on the stability of planetary motions Jacobi reduction
Planetary Birkhoff normal forms
Luigi Chierchia Gabriella Pinzari
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.
keywords: Birkhoff normal form Planetary system N-body problem Birkhoff invariants Long-time stability.
Properly-degenerate KAM theory (following V. I. Arnold)
Luigi Chierchia Gabriella Pinzari
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

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