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A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133, Milan, Italy

    Marco Sansottera & Antonio Giorgilli

  2. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Ugo Locatelli

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  1. Marco Sansottera
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  2. Ugo Locatelli
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  3. Antonio Giorgilli
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Correspondence to Marco Sansottera.

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Sansottera, M., Locatelli, U. & Giorgilli, A. A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems. Celest Mech Dyn Astr 111, 337 (2011). https://doi.org/10.1007/s10569-011-9375-x

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  • DOI: https://doi.org/10.1007/s10569-011-9375-x

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