Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T02:15:10.406Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the existence of invariant punctured tori in the planar circular restricted three-body problem

Published online by Cambridge University Press:  10 December 2009

Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The existence of transversal ejection—collision orbits in the restricted three-body problem is shown to imply, via the KAM theorem, the existence, for certain intervals of (large) values of the Jacobi constant, of an uncountable number of invariant punctured tori in the corresponding (non-compact) energy surface. The proof is based on a comparison between Levi-Civita and McGehee regularizing variables. That these transversal ejection-collision orbits do actually exist was proved in [5] in the case where one of the primaries has a small mass and the zero-mass body revolves around the other (and for all values of the Jacobi constant compatible with the existence of three connected components for the Hill region); it is proved here without any restriction on the masses, well in the spirit of Conley's thesis [3].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 63 - 72
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Birkhoff, G. D.. Sur le problème restreint des trois corps (ler mémoire). Annali Scuola Normale Superiore de Pisa, S.2 4 (1935), 267–306.Google Scholar
Chenciner, A.. Le problème de la lune et la théorie des systèmes dynamiques, lère partie. Preprint. Université Paris VII.Google Scholar
Conley, C.. On some new long periodic solutions of the plane restricted three body problem. Commun. Pure Appl. Math. XVI (1963), 449–467.10.1002/cpa.3160160405CrossRefGoogle Scholar
Devaney, R. L.. Singularities in classical mechanical systems. In Ergodic Theory and Dynamical Systems I, Proceedings Special Year, Maryland 1979–80, ed. Katok, A.. Birkhäuser, Basel (1981), 221–333.Google Scholar
Lacomba, E. & Llibre, J.. Transversal ejection-collision orbits for the restricted problem and the Hill's problem with applications. J. Differential Equations. To be published.Google Scholar
McGehee, R.. Singularities in classical celestial mechanics. Proc. ICM, Helsinki (1978), 827–834.Google Scholar
Moser, J. K.. Regularization of Kepler's problem and the averaging method on a manifold. Commun. Pure Appl. Math. XXIII (1970), 609–636.10.1002/cpa.3160230406CrossRefGoogle Scholar
Sanders, J. A.. Melnikov's method and averaging. Celestial Mech. 28 (1982), 171–181.10.1007/BF01230669CrossRefGoogle Scholar
Siegel, C. L. & Moser, J. K.. Lectures on Celestial Mechanics. Springer, New York (1971).CrossRefGoogle Scholar