Whiskered Parabolic Tori in the Planar \((n+1)\)-Body Problem

  • Inmaculada Baldomá
  • Ernest Fontich
  • Pau MartínEmail author


The planar \((n+1)\)-body problem models the motion of \(n+1\) bodies in the plane under their mutual Newtonian gravitational attraction forces. When \(n\ge 3\), the question about final motions, that is, what are the possible limit motions in the planar \((n+1)\)-body problem when \(t\rightarrow \infty \), ceases to be completely meaningful due to the existence of non-collision singularities. In this paper we prove the existence of solutions of the planar \((n+1)\)-body problem which are defined for all forward time and tend to a parabolic motion, that is, that one of the bodies reaches infinity with zero velocity while the rest perform a bounded motion. These solutions are related to whiskered parabolic tori at infinity, that is, parabolic tori with stable and unstable invariant manifolds which lie at infinity. These parabolic tori appear in cylinders which can be considered “normally parabolic”. The existence of these whiskered parabolic tori is a consequence of a general theorem on parabolic tori developed in this paper. Another application of our theorem is a conjugation result for a class of skew product maps with a parabolic torus with its normal form generalizing results of Takens (Ann Inst Fourier 23(2):163–195, 1973), and Voronin (Funktsional Anal i Prilozhen 15(1):1–17, 96, 1981).



The authors thank the anonymous referees for their helpful comments.

I.B and P.M. have been partially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan Grant 2017-SGR-1049. The work of E.F. has been partially supported by the Spanish Government grant MTM2016-80117-P (MINECO/FEDER, UE) and the Catalan Government grant 2017-SGR-1374. Also all authors have been partially supported by the Maria de Maeztu project MDM-2014-044.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de Matemàtiques, Barcelona Graduate School of Mathematics (BGSMath)Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB), Barcelona Graduate School of Mathematics (BGSMath)Universitat de Barcelona (UB)BarcelonaSpain
  3. 3.Departament de Matemàtiques, Barcelona Graduate School of Mathematics (BGSMath)Universitat Politècnica de CatalunyaBarcelonaSpain

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