Isolating blocks near the collinear relative equilibria of the three-body problem

Author:
Richard Moeckel

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 4395-4425

MSC (2000):
Primary 70F10, 70F15, 37N05

DOI:
https://doi.org/10.1090/S0002-9947-04-03418-X

Published electronically:
January 23, 2004

MathSciNet review:
2067126

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body 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 three-body 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,*Sun-perturbed earth-to-moon transfers with ballistic capture*, Jour. of Guidance, Control and Dynamics,**16**(1993) 770-775.**3.**A. Albouy and A. Chenciner,*Le probléme des n corps et les distances mutuelles*, Inv. Math.,**131**(1998) 151-184. MR**98m:70017****4.**C. C. Conley,*Low energy transit orbits in the restricted three-body problem*, SIAM J. Appl. Math.,**16, 4**(1968) 732-746. 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) 35-60. MR**43:5551****7.**R. W. Easton,*Existence of invariant sets inside a submanifold convex to the flow*, JDE,**7**(1970) 54-68. MR**40:2996****8.**R. W. Easton,*Some qualitative aspects of three-body flow*, in Dynamical Systems, vol. 2, An International Symposium, Academic Press (1976) 1-6. 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) 129-146. 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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
70F10,
70F15,
37N05

Retrieve articles in all journals with MSC (2000): 70F10, 70F15, 37N05

Additional Information

**Richard Moeckel**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
rick@math.umn.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03418-X

Keywords:
Celestial mechanics,
central configurations,
three-body 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