论文标题
1+N辅助问题中的对称性和不对称性
Symmetry and Asymmetry in the 1+N Coorbital Problem
论文作者
论文摘要
当群众以外的所有群众均为零时，牛顿$ n $ n $ n $ n $ body问题的相对平衡变得在极限范围内。我们证明了有关辅助相对平衡的各种结果，并强调了构型的对称性与质量对称性之间的关系，或者缺乏对称性。我们证明，在$ n = 4 $中，$ n = 6 $，$ n = 8 $ newtonian coorbital问题存在对称的相对平衡，具有不对称的正质量。该结果可以推广到其他均质电位，我们猜测对于较大的无限质量数量的相似结果。我们证明，$ 1+4 $和$ 1+5 $ coorbital问题中的群众的某些平等性暗示着一类凸相的相对平衡的对称性。我们还证明，对称$ 1+5 $问题的最多有一个凸中心配置。
The relative equilibria of planar Newtonian $N$-body problem become 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.
