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Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

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Abstract

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We consider only the circular case (e = 0) but under the action of a broader family of singular potentials: α-homogeneous potentials, for \(\alpha \in (0, 2)\), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter α and the mass parameter β, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y. Long, X. Hu and S. Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto, 10, 10123, Turin, Italy

    Vivina Barutello

  2. Dipartimento di Scienze Matematiche “J.-L. Lagrange” (DISMA), Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

    Riccardo D. Jadanza

  3. Institute for Scientific Interchange (ISI Foundation), Via Alassio, 11/C, 10126, Torino, Italy

    Riccardo D. Jadanza

  4. Dipartimento di Scienze Agrarie, Forestali e Alimentari (DISAFA), Università degli Studi di Torino, Largo Paolo Braccini, 2, 10095, Grugliasco, TO, Italy

    Alessandro Portaluri

Authors
  1. Vivina Barutello
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  2. Riccardo D. Jadanza
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  3. Alessandro Portaluri
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Corresponding author

Correspondence to Vivina Barutello.

Additional information

Communicated by P. Rabinowitz

The authors are partially supported by the project ERC Advanced Grant 2013 no. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”.

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Barutello, V., Jadanza, R.D. & Portaluri, A. Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory. Arch Rational Mech Anal 219, 387–444 (2016). https://doi.org/10.1007/s00205-015-0898-2

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  • DOI: https://doi.org/10.1007/s00205-015-0898-2

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