Abstract
We investigate specific homothetic solutions of then-body problem which both begin and end in a simultaneous collision of all of the particles. Under a suitable change of variables, these solutions become heteroclinic orbits, i.e., they lie in the intersection of the stable and unstable manifolds of distinct equilibrium points. Our main result is that these manifolds intersect transversely along these orbits. This proves that the homothetic solutions are structurally stable.
Similar content being viewed by others
References
Abraham, R. and Robbin, J.: 1967,Transversal Mappings and Flows, W. A. Benjamin, New York.
Kuz'mina, R. P.: 1977,Soviet Math. Dokl. 18, 818.
McGehee, R.: 1974,Invent. Math. 27, 191.
Milnor, J.: 1963,Morse Theory, Ann. Math. Stud., No. 41, Princeton University Press, Princeton, N.J.
Palmore, J.: 1973,Bull. Am. Math. Soc. 79, 904.
Palmore, J.: 1975,Bull. Am. Math. Soc. 81, 489.
Saari, D. G.: 1973,Trans. Am. Math. Soc. 181, 351.
Smale, S.: 1965,Differential and Combinatorial Topology, Princeton University Press, Princeton, N.J.
Smale, S.: 1970,Invent. Math. 11, 45.
Wintner, A.: 1952,The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, N.J.
Author information
Authors and Affiliations
Additional information
Partially supported by NSF Grant MCS 77-00430.
Rights and permissions
About this article
Cite this article
Devaney, R.L. Structural stability of homothetic solutions of the collinearn-body problem. Celestial Mechanics 19, 391–404 (1979). https://doi.org/10.1007/BF01231016
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01231016