Corrigendum to: Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

Abstract

The main result of the paper [BCV] Elliptic two-dimensional invariant tori for the planetary three-body problem by the authors, published in Arch. Ration. Mech. Anal. 170, 91–135 (2003), namely, Theorem 1.2, exploits a classical result (Theorem 1.1) essentially due to Delaunay and Poincare. In Appendix C of [BCV] a detailed proof of Theorem 1.1 is discussed. However, this proof contains a flaw. In fact, the statement following equation (C.44) in Proposition C.3, Section C.2.2, Appendix C is not correct. The Z∗ j ’s (called in [BCV] “Poincare integrals”) are not global integrals. Therefore, Section C.2.2 requires amendment. What now follows, replaces Section C.2.2 in [BCV], correcting the above mentioned flaw. Notations are as in [BCV]. Recall that in [BCV] we consider the 10-dimensional invariant symplectic manifold, Mver, defined by taking the non-vanishing total angular momentum to be vertical, i.e., C:=C(1) + C(2) = ck3 where C denotes the angular momentum of the ith planet with respect to the origin of an inertial heliocentric frame {k1, k2, k3} and c ∈ R\{0}; compare (C.30) and (C.31). Recall, also, that in [BCV] we use the “osculating Poincare variables” ( ∗, η∗, p∗, λ∗, ξ∗, q∗) as described in Section C.2.1 of [BCV], where ∗ = ( ∗1, ∗2), η∗ = (η∗ 1, η∗ 2), etc., and the index i = 1, 2 refers to the two-body system ith planet-star. Finally, recall that −ζ ∗ i = θi is the longitude of the ith node on the plane {k1, k2}, while −Z∗ i = i = C · k3. Let us now describe, analytically, Mver. From the definition of the Delaunay variables (Li, i,Gi, gi, i, θi) (compare to Section C.1.1) it follows that the relations defining Mver, i.e., C · k1 = 0 = C · k2, are equivalent to ζ ∗ 2 − ζ ∗ 1 = π and G1 − 1 = G2 − 2. Therefore, since Gi = ∗i − H ∗ i , with H ∗ i :=(η∗ i 2 + ξ∗ i )/2 (compare to (C.5), (C.15) and (C.17)), we find