On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems

Abstract

We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic Keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.