Abstract
The main goal of this paper is to show that the elliptic restricted three-body problem has ejection-collision orbits when the mass parameter µ is small enough. We make use of the ‘blow up’ techniques. Moreover, we describe the global flow of the elliptic problem when µ = 0 taking into account the singularities due to collision and to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R. and Marsden, J. E.: 1978, Foundations of Mechanics, Benjamin, Reading, Massachusetts.
Chenciner, A. and Llibre, J.: 1988, ‘A note on the existence of invariant punctured tori in the planar circular restricted three-body problem’, Ergodic Theory and Dyn. Syst. 8 *, 63–72.
Contopoulos, G.: 1967, ‘Integrals of motion in the elliptic restricted Three-Body Problems’, Astron. J. 72,669–673.
Devaney, R. L.: 1978, ‘Collision Orbits in the Anisotropic Kepler Problem’, Inventiones Math. 45, 221–251.
Devaney, R. L.: 1981, ‘Singularities in Classical Mechanical Systems’, in Ergodic Theory and Dynamical Systems 1, Proceedings Special, Maryland 1979–80, A. Kato (Ed.), pp. 211–333, Birkhäuser, Basel.
Guillemin, V. and Pollach, L: 1974, Differential Topology, Prentice-Hall.
Hirsh, M., Pugh, C. and Shub, M.: 1977, ‘Invariant Manifolds’, Lecture Notes in Mathematics, Vol. 583, Springer Verlag, Berlin.
Lacomba, E. A. and Llibre, J.: 1988, ‘Transversal ejection-collision orbits for the restricted problem and the Hill's problem with applications, J. Differential Equations 74, 69–85.
Llibre, J. and Simo, C.: 1980, ‘Oscillatory solutions in the planar restricted three-body problem’, Math. Ann 248, 157–184.
Llibre, J.: 1982, ‘On the restricted three-body problem when the mass parameter is small’, Celest. Mech. 28,83–105.
Llibre, J.. and Martiniez Alfaro J.: 1985, ‘Ejection and collision orbits of the spatial restricted three-body problem’, Celest. Mech. 35, 113–128.
Llibre, J. and Pinyol, C.: 1989, ‘Transversal ejection-collision orbits for the restricted three-body problem’, Proceedings of European Conference on Iteration Theory (ECIT 87), World Scientific, Singapore, 263–270.
McGehee, R.: 1974, ‘Triple collision in the collinear three-body problem’, Inventiones Math. 27, 191–227.
Pinyol, C.: 1987, ‘Estudi qualitatiu d'alguns problemes restringits de 3 ó 4 cossos’, Ph.D., Universitat Autonoma de Barcelona.
Siegel, C. L. and Moser, J. K.: 1971, Lectures on Celestial Mechanics, Springer-Verlag, Berlin.
Stiefel, E. L. and Scheifele, G.: 1971, Linear and Regular Celestial Mechanics, Springer-Verlag, Berlin.
Szebehely, V.: 1967, Theory of Orbits, Academic Press, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Llibre, J., Piñol, C. On the elliptic restricted three-body problem. Celestial Mech Dyn Astr 48, 319–345 (1990). https://doi.org/10.1007/BF00049388
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00049388