Abstract
We complete Mc Gehee's picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds. representing now the asymptotic behavior at infinity.
It is necessary to treat each caseh=0,h>0 orh<0 separately. In the first case, we repeat the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is radically different in theh>0 case compared with the zero energy case. In the caseh<0 we have many infinity manifold components. and the general situation is not quite well understood.
Finally, our results forh≥0 are shown to be valid for general homogeneous potentials.
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References
Devaney, R.: 1980,J. Diff. Geom 15, 285–305.
Lacomba, E. A., and Losco, L.: 1980,Bull. Am. Math. Soc. (New Series)3, 710–714.
Marchal, C. and Losco, L.: 1980,Astron. Astrophys. 84, 1–6.
Marchal, C. and Saari, D. G.: 1976,J. Diff. Eq. 20, 150–186.
McGehee, R.: 1974,Inventiones Math. 27, 191–227.
Simó, C.: 1981, in R. L. Devaney and Z. H. Nitecki (eds.),Classical Mechanics and Dynamical Systems, Marcel Dekker, New York, p. 203.
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The research conducted in this paper has been partially supported by CONACYT (México), under grant PCCBNAL 790178.
Partially supported by an Ajut a l'Investigacio of the University of Barcelona.
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Lacomba, E.A., Simó, C. Boundary manifolds for energy surfaces in celestial mechanics. Celestial Mechanics 28, 37–48 (1982). https://doi.org/10.1007/BF01230658
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DOI: https://doi.org/10.1007/BF01230658