Construction of the Normal Form for Elliptic Tori in Planetary Systems. Part I: Numerical Validation of the Semi-Analytic Algorithm

Details Construction of the Normal Form for Elliptic Tori in Planetary Systems. Part I: Numerical Validation of the Semi-Analytic Algorithm Construction of the Normal Form for Elliptic Tori in Planetary Systems. Part I: Numerical Validation of the Semi-Analytic Algorithm

2010-10-13

arxiv.org/abs/1010.2617v1

Mathematics

Dynamical Systems

28 pages, 4 figures

Scientific paper

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.

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