Journal of Modern Dynamics (JMD)

Planetary Birkhoff normal forms

Pages: 623 - 664, Issue 4, October 2011      doi:10.3934/jmd.2011.5.623

       Abstract        References        Full Text (532.0K)       Related Articles       

Luigi Chierchia - Dipartimento di Matematica, Università "Roma Tre", Largo S. L. Murialdo 1, 00146 Roma, Italy (email)
Gabriella Pinzari - Dipartimento di Matematica, Università “Roma Tre”, Largo S. L. Murialdo 1, I-00146 Roma, Italy (email)

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.

Keywords:  Planetary system, Birkhoff normal form, Birkhoff invariants, N-body problem, Long-time stability.
Mathematics Subject Classification:  70F10, 70K45, 70F15, 70F07, 70E55, 34C20, 34D10.

Received: September 2010;      Revised: October 2011;      Available Online: March 2012.