A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

Abstract

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.

31 pages, 4 figures

References in corpus (4)

• Kolmogorov and Nekhoroshev theory for the problem of three bodies
• On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system
• On the Kolmogorov set for Many-Body Problems
• Su un'estensione della teoria di Lagrange per i moti secolari

Cited by in corpus (16)

• Global structure of regular tori in a generic 4D symplectic map
• Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
• On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems
• Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties
• Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance
• Trojan dynamics well approximated by a new Hamiltonian normal form
• Rigorous estimates for the relegation algorithm
• 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
• On the continuation of degenerate periodic orbits via normal form: lower dimensional resonant tori
• Normal form for lower dimensional elliptic tori in Hamiltonian systems
• On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori
• On the relativistic Lagrange-Laplace secular dynamics for extrasolar systems
• Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms
• The construction of averaged planetary motion theory by means of computer algebra system Piranha
• Quasi-periodic motions in a special class of dynamical equations with dissipative effects: a pair of detection methods