Abstract
Birkhoff normal forms for the (secular) planetary problem are investigated. Existence and uniqueness is discussed and it is shown that the classical Poincare 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 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.
Highlights
Birkhoff normal forms for the planetary problem are investigated
A second remarkable feature of the planetary system is that the secular Hamiltonian has an elliptic equilibrium around zero inclinations and eccentricities
Secular Birkhoff invariants are intimately related to the existence of maximal and lower-dimensional KAM tori2, or, as we will show below (§ 6), one can infer long-time stability for the “secular actions”
Summary
After the symplectic reduction of the linear momentum, the (1 + n)-body problem with masses m0, μm1, · · · , μmn (0 < μ 1) is governed by the 3ndegrees-of-freedom Hamiltonian. Ps being homogeneous polynomial in r of order s, parameterized by Λ Such normal form is unique up to symplectic transformations Φ which leave the Λ’s fixed and with the z-projection independent of l and close to the identity in w, i.e.,. (ii) Theorem 2.1 depends strongly on the rotational invariance of the Hamiltonian (2.1), that is, on the fact that such Hamiltonian commutes with the three components of the angular momentum C To exploit explicitly such invariance, we shall use a set of symplectic variables (“RPS variables”), introduced in [6] (in order to describe the symplectic structure of the planetary N-body problem and to check KAM nondegeneracies). The RPS variables have a cyclic couple ((pn, qn) below), which foliates the phase space into symplectic leaves (the sets M(6pnn−,q2n ) in (3.13) below), on which the planetary Hamiltonian keeps the same form. The proof is based on the remarkable link between RPS and Poincaré variables, described (see Theorem 3.2)
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