Equilibrium and stability in Robe’s restricted problem with a line segment and modified potential

Abstract

In mathematics, particularly in dynamical systems theory, equilibrium points and their stability are of critical importance, as they represent the steady-state solutions of a system. This study examines the equilibrium points and their stability in a modified formulation of a well known dynamical system: the Robe’s problem. Unlike the original Robe problem where the primary \(m_{2}\) is treated as a point mass, we have modeled \(m_{2}\) as a line segment. The line segment is assumed to produce modified Newtonian potential instead of classical Newtonian potential. The introduction of a modified Newtonian potential leads to substantial changes in the system’s dynamics when compared to the classical Robe’s problem with line segment. Using analytical techniques, we show that the modified system admits only a single collinear equilibrium point, in contrast to the classical case, which allows infinite equilibrium points, including non-collinear (degenerate equilibrium set) and out-of-plane equilibrium points. A linear stability analysis of this equilibrium point indicates marginal stability, with all eigenvalues lying purely on the imaginary axis. This suggests that small perturbations neither grow exponentially nor decay, indicating the absence of inherent damping or restoring mechanisms. Overall, the findings highlight how departures from point-mass assumptions and classical Newtonian potentials can significantly influence the qualitative dynamics of Robe’s restricted three-body system by breaking the degeneracy of equilibrium configuration.