Planetary Birkhoff normal forms

Birkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown the classical Poincare variables and the RPS-variables (introduced in [6]), after a trivial lift, lead to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincare 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.