Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result
Gabriella Pinzari
Electronic Research Announcements 2015, 22(0): 55-75 doi: 10.3934/era.2015.22.55
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
Journal of Modern Dynamics 2011, 5(4): 623-664 doi: 10.3934/jmd.2011.5.623
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
Discrete & Continuous Dynamical Systems - S 2010, 3(4): 545-578 doi: 10.3934/dcdss.2010.3.545
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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