Symmetry and Asymmetry in the 1 + N Coorbital Problem

Yiyang Deng; Marshall Hampton; Zhiqiang Wang

doi:10.1137/22M1475053

Symmetry and Asymmetry in the 1 + N Coorbital Problem

Yiyang Deng, Marshall Hampton, Zhiqiang Wang

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

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)
Pages (from-to)	2080-2095
Number of pages	16
Journal	SIAM Journal on Applied Dynamical Systems
Volume	21
Issue number	3
DOIs	https://doi.org/10.1137/22M1475053
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 (dengyiyang@126.com). ‡Department of Mathematics and Statistics, University of Minnesota, Duluth, Duluth, MN 55812 USA (mhampton@d.umn.edu). §College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China (wzq2015@cqu.edu.cn).

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

Keywords

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

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title = "Symmetry and Asymmetry in the 1 + N Coorbital Problem",

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.",

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author = "Yiyang Deng and Marshall Hampton and Zhiqiang Wang",

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 (dengyiyang@126.com). ‡Department of Mathematics and Statistics, University of Minnesota, Duluth, Duluth, MN 55812 USA (mhampton@d.umn.edu). §College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China (wzq2015@cqu.edu.cn). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

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Symmetry and Asymmetry in the 1 + N Coorbital Problem

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