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

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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.

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Notes

  1. Precisely, \(\mathcal{G}_{\varrho }=\big \{z\in \mathbb {C}^{n_1}:\max _{1\le j\le n_1}|z_j|<\varrho \big \}\), \(\mathbb {T}^{n_1}_{\sigma }=\big \{q\in \mathbb {C}^{n_1}:\mathrm{Re}q_j\in \mathbb {T}, \max _{1\le j\le n_1}|\mathrm{Im}q_j|< \sigma \big \}\), \(\mathcal{B}_R=\{z\in \mathbb {C}^{2n_2}: \max _{1\le j\le 2n_2} |z_j|<R\,\}\) and \(\mathcal{W}_h = \big \{z\in \mathbb {C}^{n_1}: \exists \ \omega \in \mathcal{W}, \max _{1\le j\le n_1}|z_j-\omega _j|<h\big \}\).

  2. First, let us remark that \(\varepsilon ^{\star }_{\mathrm{an}}=\mathcal{O}(e^{-6\varGamma })\) in view of the definitions in (61). By definitions 2 and 3 we get that \(\varGamma =\sum \limits _{r\ge 1} [-\log \gamma +\tau \log (rK)]/[r(r+1)]\,\). Therefore, \(e^{-\varGamma }=\mathcal{O}(\gamma )\,\), as one can easily verify that \(\sum \limits _{r\ge 1} 1/[r(r+1)]=1\,\). For this purpose, it is enough to check by induction that \(\sum \limits _{r=1}^{s} 1/[r(r+1)]=s/(s+1)\) for \(s\ge 1\,\).

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Acknowledgments

A. G. and U. L. have been partially supported by the research program “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”, PRIN 2010JJ4KPA_009, financed by MIUR. U. L. has been partially supported also by Marie Curie ITN “Astronet II”, EC contract PITN-GA-2011-289240. The work of M. S. has been supported by an FSR Incoming Post-doctoral Fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission.

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Authors and Affiliations

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

    Antonio Giorgilli

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

    Ugo Locatelli

  3. Département de Mathématique and NAXYS, Université de Namur, Rempart de la Vierge 8, 5000 , Namur, Belgium

    Marco Sansottera

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  1. Antonio Giorgilli
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Correspondence to Marco Sansottera.

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Giorgilli, A., Locatelli, U. & Sansottera, M. On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems. Celest Mech Dyn Astr 119, 397–424 (2014). https://doi.org/10.1007/s10569-014-9562-7

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