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

Autor: Riccardo D. Jadanza, Alessandro Portaluri, Vivina Barutello
Rok vydání: 2015
Předmět:
Zdroj: Archive for Rational Mechanics and Analysis. 219:387-444
ISSN: 1432-0673
0003-9527
DOI: 10.1007/s00205-015-0898-2
Popis: 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 $\beta$ 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: $\alpha$-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 $\alpha$ and the mass parameter $\beta$, 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.
Comment: 48 pages, 13 figures, minor revisions
Databáze: OpenAIRE
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