Isolating blocks near the collinear relative equilibria of the threebody problem
Author:
Richard Moeckel
Translated by:
Journal:
Trans. Amer. Math. Soc. 356 (2004), 43954425
MSC (2000):
Primary 70F10, 70F15, 37N05
Published electronically:
January 23, 2004
MathSciNet review:
2067126
Fulltext PDF Free Access
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Abstract: The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian threebody problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted threebody problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.
 1.
D. Appleyard, thesis, University of Wisconsin, (1968).
 2.
E. Belbruno and J. Miller, Sunperturbed earthtomoon transfers with ballistic capture, Jour. of Guidance, Control and Dynamics, 16 (1993) 770775.
 3.
Alain
Albouy and Alain
Chenciner, Le problème des 𝑛 corps et les distances
mutuelles, Invent. Math. 131 (1998), no. 1,
151–184 (French). MR 1489897
(98m:70017), http://dx.doi.org/10.1007/s002220050200
 4.
C.
C. Conley, Low energy transit orbits in the restricted threebody
problem, SIAM J. Appl. Math. 16 (1968),
732–746. MR 0233535
(38 #1856)
 5.
Charles
Conley, Isolated invariant sets and the Morse index, CBMS
Regional Conference Series in Mathematics, vol. 38, American
Mathematical Society, Providence, R.I., 1978. MR 511133
(80c:58009)
 6.
C.
Conley and R.
Easton, Isolated invariant sets and isolating
blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61. MR 0279830
(43 #5551), http://dx.doi.org/10.1090/S00029947197102798301
 7.
Robert
W. Easton, On the existence of invariant sets inside a submanifold
convex to a flow, J. Differential Equations 7 (1970),
54–68. MR
0249755 (40 #2996)
 8.
Robert
W. Easton, Some qualitative aspects of the threebody flow,
Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I.,
1974) Academic Press, New York, 1976, pp. 1–6. MR 0665443
(58 #32196)
 9.
Wang
Sang Koon, Jerrold
E. Marsden, Shane
D. Ross, and Martin
W. Lo, Constructing a low energy transfer between Jovian
moons, Celestial mechanics (Evanston, IL, 1999) Contemp. Math.,
vol. 292, Amer. Math. Soc., Providence, RI, 2002,
pp. 129–145. MR 1884895
(2002m:70040), http://dx.doi.org/10.1090/conm/292/04919
 10.
J. L. Lagrange, Ouvres, vol 6, 272.
 11.
R. Moeckel, Sturm's algorithm and isolating blocks, to appear in JSC.
 12.
G.
Gómez, J.
Llibre, R.
Martínez, and C.
Simó, Dynamics and mission design near libration points.
Vol. I, World Scientific Monograph Series in Mathematics, vol. 2,
World Scientific Publishing Co., Inc., River Edge, NJ, 2001. Fundamentals:
the case of collinear libration points; With a foreword by Walter Flury. MR 1867240
(2003c:70041a)
G.
Gómez, À.
Jorba, C.
Simó, and J.
Masdemont, Dynamics and mission design near libration points. Vol.
III, World Scientific Monograph Series in Mathematics, vol. 4,
World Scientific Publishing Co., Inc., River Edge, NJ, 2001. Advanced
methods for collinear points. MR 1878993
(2003c:70041c)
 13.
B.
L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar
Publishing Co., New York, N. Y., 1949. Translated from the second revised
German edition by Fred Blum; With revisions and additions by the author. MR 0029363
(10,587b)
 14.
Aurel
Wintner, The Analytical Foundations of Celestial Mechanics,
Princeton Mathematical Series, v. 5, Princeton University Press, Princeton,
N. J., 1941. MR
0005824 (3,215b)
 1.
 D. Appleyard, thesis, University of Wisconsin, (1968).
 2.
 E. Belbruno and J. Miller, Sunperturbed earthtomoon transfers with ballistic capture, Jour. of Guidance, Control and Dynamics, 16 (1993) 770775.
 3.
 A. Albouy and A. Chenciner, Le probléme des n corps et les distances mutuelles, Inv. Math., 131 (1998) 151184. MR 98m:70017
 4.
 C. C. Conley, Low energy transit orbits in the restricted threebody problem, SIAM J. Appl. Math., 16, 4 (1968) 732746. MR 38:1856
 5.
 C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, 38, American Mathematical Society (1978). MR 80c:58009
 6.
 C. C. Conley and R. W. Easton, Isolated invariant sets and isolating blocks, Trans. AMS, 158, 1 (1971) 3560. MR 43:5551
 7.
 R. W. Easton, Existence of invariant sets inside a submanifold convex to the flow, JDE, 7 (1970) 5468. MR 40:2996
 8.
 R. W. Easton, Some qualitative aspects of threebody flow, in Dynamical Systems, vol. 2, An International Symposium, Academic Press (1976) 16. MR 58:32196
 9.
 W. S. Koon, et al., Constructing a low energy transfer between Jovian moons, in Celestial Mechanics, Contemporary Mathematics, vol. 292, American Mathematical Society (2002) 129146. MR 2002m:70040
 10.
 J. L. Lagrange, Ouvres, vol 6, 272.
 11.
 R. Moeckel, Sturm's algorithm and isolating blocks, to appear in JSC.
 12.
 C. Simo, et al., Dynamics and Mission Design Near Libration Points, vols. I, III, World Scientific (2001). MR 2003c:70041a, MR 2003c:70041c
 13.
 B. L. van der Waerden, Modern Algebra, Ungar, New York (1949). MR 10:587b
 14.
 A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series 5, Princeton University Press, Princeton, NJ (1941). MR 3:215b
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Additional Information
Richard Moeckel
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
rick@math.umn.edu
DOI:
http://dx.doi.org/10.1090/S000299470403418X
PII:
S 00029947(04)03418X
Keywords:
Celestial mechanics,
central configurations,
threebody problem
Received by editor(s):
December 11, 2002
Received by editor(s) in revised form:
May 7, 2003
Published electronically:
January 23, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
