## Abstract

The relative equilibria of the planar Newtonian N-body problem becomes coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the N = 4, N = 6, and N = 8 Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hold for larger even numbers of infinitesimal masses. We prove that some equalities of the masses in the 1+4 and 1+5 coorbital problems imply symmetry of a class of convex relative equilibria. We also prove there is at most one convex central configuration of the symmetric 1 + 5 problem.

Original language | English (US) |
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Pages (from-to) | 2080-2095 |

Number of pages | 16 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 21 |

Issue number | 3 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Funding Information:∗Received by the editors January 31, 2022; accepted for publication (in revised form) May 16, 2022; published electronically August 8, 2022. https://doi.org/10.1137/22M1475053 Funding: The first author was partially supported by the Science and Technology Research Program of Chongqing Municipal Education Commission grant KJQN201800815, the Research Program of CTBU grant 1952040, and the Program for the Introduction of High-Level Talents of CTBU grant 1856010. †College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China ([email protected]). ‡Department of Mathematics and Statistics, University of Minnesota, Duluth, Duluth, MN 55812 USA ([email protected]). §College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China ([email protected]).

Publisher Copyright:

© 2022 Society for Industrial and Applied Mathematics.

## Keywords

- N-body problem
- celestial mechanics
- central configurations
- coorbital problem
- relative equilibria