No infinite spin for planar total collision
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- by Richard Moeckel and Richard Montgomery;
- J. Amer. Math. Soc. 38 (2025), 225-241
- DOI: https://doi.org/10.1090/jams/1044
- Published electronically: January 18, 2024
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Abstract:
The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality.References
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Bibliographic Information
- Richard Moeckel
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 125920
- Email: rick@math.umn.edu
- Richard Montgomery
- Affiliation: Dept. of Mathematics, University of California, Santa Cruz, California
- MR Author ID: 126555
- ORCID: 0000-0001-5293-6837
- Email: rmont@ucsc.edu
- Received by editor(s): February 13, 2023
- Received by editor(s) in revised form: October 25, 2023
- Published electronically: January 18, 2024
- © Copyright 2024 American Mathematical Society
- Journal: J. Amer. Math. Soc. 38 (2025), 225-241
- MSC (2020): Primary 37N05, 70F10, 70F15, 70F16, 70G40, 70G60
- DOI: https://doi.org/10.1090/jams/1044