报告题目：On the vertical librations of the ring problem of the (N+1) bodies with a central normal ellipsoid

报 告 人：徐兴波 副教授（淮阴工学院）

报告时间：2025年12月5日（星期五）9:00—10:00

报告地点：数学科学学院115（大报告厅）

校内联系人：赵磊 教授     联系方式：84708351-8625

报告摘要：Maxwell's ring model, which is a central configuration of the $(N+1)$-body problem with $N$ small equal satellites, was used to show the stability of the rings of Saturn by Maxwell(1859). Meyer & Schmidt(1993), Moeckel(1994), and so on, studied the related problem successively.

In this talk, we consider the vertical librations of the uniform ring in the $(N+1)$-body problem with a central normal ellipsoid. The ring lies in the equatorial plane of the central primary. There are two kinds of periodic vertical libration, which are non-alternating and alternating cases. The gravitational field of the central primary is highly symmetric and all odd zonal harmonic terms are neglected. The Hamiltonians can be reduced to a simple form with two degrees of freedom. The Hamiltonians are expanded near the planar regular $N$-gon configuration. The formally normal form up to the quadratic form is achieved by the Lie transformation technique. The coefficients are shown to be convergent when $|J_{2n}|<10^{-(n-1)} J_{2}$ for $n>1$. The KAM stability condition is enhanced by the $J_2$ terms, according to Arnold's stability theorem. In addtion, some periodic orbits are also calculated and studied.