Planetary Birkhoff normal forms

Luigi Chierchia; Gabriella Pinzari

doi:10.3934/jmd.2011.5.623

Article Contents

2011, Volume 5, Issue 4: 623-664. Doi: 10.3934/jmd.2011.5.623

This issue Previous Article Next Article Spectral analysis of the transfer operator for the Lorentz gas

Planetary Birkhoff normal forms

Luigi Chierchia^1, and
Gabriella Pinzari^2,

1.
Dipartimento di Matematica, Università "Roma Tre", Largo S. L. Murialdo 1, 00146 Roma
2.
Dipartimento di Matematica, Università “Roma Tre”, Largo S. L. Murialdo 1, I-00146 Roma

Received: September 2010

Revised: October 2011

Published: March 2012

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Abstract

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.

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Mathematics Subject Classification: 70F10, 70K45, 70F15, 70F07, 70E55, 34C20, 34D10.

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