Bifurcations of the Hill’s region in the three body problem
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- by Christopher K. McCord PDF
- Proc. Amer. Math. Soc. 127 (1999), 2135-2142 Request permission
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.References
- Alain Albouy, Integral manifolds of the $n$-body problem, Invent. Math. 114 (1993), no. 3, 463–488. MR 1244910, DOI 10.1007/BF01232677
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Christopher K. McCord, Kenneth R. Meyer, and Quidong Wang, The integral manifolds of the three body problem, Mem. Amer. Math. Soc. 132 (1998), no. 628, viii+90. MR 1407897, DOI 10.1090/memo/0628
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
- Donald G. Saari, From rotations and inclinations to zero configurational velocity surfaces. II. The best possible configurational velocity surfaces, Celestial Mech. 40 (1987), no. 3-4, 197–223. MR 938403, DOI 10.1007/BF01235841
- Donald G. Saari, From rotations and inclinations to zero configurational velocity surfaces. I. A natural rotating coordinate system, Celestial Mech. 33 (1984), no. 4, 299–318. MR 777381, DOI 10.1007/BF01241046
- Donald G. Saari, Symmetry in $n$-particle systems, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 23–42. MR 986255, DOI 10.1090/conm/081/986255
- C. Simo, El conjunto de bifurcacion en el problema espacial de tres cuerpos, Acta I Asamblea Nacional de Astronomia y Astrofisica (1975), 211-217, Instituto de Astrofisica, Univ. de la Laguna, Spain, 211-217.
Additional Information
- Christopher K. McCord
- Affiliation: Institute for Dynamics, Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
- Email: chris.mccord@uc.edu
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1487328