On the Vertical Libration of Maxwell’s Ring Configuration with a Manev Potential

We discuss Maxwell’s ring model with a quasi-homogeneous potential of the dominant mass, and specially study the vertical libration of Maxwell’s ring configuration with a Manev potential. There are two kinds of periodic vertical libration, which are non-alternating and alternating cases. The Hamiltonian is highly symmetric and can be reduced to a simple form with two degrees of freedom. Then the Hamiltonian is expanded near the planar circular relative equilibria. By the Lie transformation technique, the birkhoff norm form up to the fourth order is calculated. When the mass ratio between the ring and the dominant mass, and the coefficient of the second term in a Manev potential are both small enough, the nonlinear stability for the vertical libration of Maxwell’s ring configuration with a Manev potential is then proved according to Arnold’s stability theorem.