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Bifurcations of the Hill's region
in the three body problem


Author: Christopher K. McCord
Journal: Proc. Amer. Math. Soc. 127 (1999), 2135-2142
MSC (1991): Primary 70F07; Secondary 57Q60, 58F14
DOI: https://doi.org/10.1090/S0002-9939-99-04755-3
Published electronically: March 3, 1999
MathSciNet review: 1487328
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Abstract: In the spatial three body problem, the topology of the integral manifolds $\mathfrak{M}(c,h)$ (i.e. the level sets of energy $h$ and angular momentum $c$, as well as center of mass and linear momentum) and the Hill's regions $\mathfrak{H}(c,h)$ (the projection of the integral manifold onto position coordinates) depends only on the quantity $\nu = h|c|^2.$ It was established by Albouy and McCord-Meyer-Wang that, for $h < 0$ and $c \neq 0$, there are exactly eight bifurcation values for $\nu$ at which the topology of the integral manifold changes. It was also shown that for each of these values, the topology of the Hill's region changes as well. In this work, it is shown that there are no other values of $\nu$ for which the topology of the Hill's region changes. That is, a bifurcation of the Hill's region occurs if and only if a bifurcation of the integral manifold occurs.


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Additional Information

Christopher K. McCord
Affiliation: Institute for Dynamics, Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: chris.mccord@uc.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04755-3
Received by editor(s): June 11, 1997
Received by editor(s) in revised form: October 14, 1997
Published electronically: March 3, 1999
Additional Notes: The author was supported in part by grants from the National Science Foundation and the Charles Phelps Taft Memorial Fund.
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society

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