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

Abstract

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's 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-analytical work in our previous article (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.

45 pages

Cited by in corpus (5)

• Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
• Elliptic tori in FPU non-linear chains with a small number of nodes
• Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system
• Improved convergence estimates for the Schröder-Siegel problem
• Secular orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedæ system