Classical orbits in an inverse-square cylindrical potential: closed-form analysis and applications

Abstract We study classical motion in the inverse-square cylindrical potential U ( ρ ) ∝ − 1/ ρ 2 . After the two-body reduction, a single dimensionless parameter η —combining interaction strength and axial angular momentum—fully sets the dynamics. This splits the motion into three regimes. For η < 1 the motion is unbound: trajectories have a unique closest approach and an analytic deflection that depends only on the impact parameter. At the threshold η = 1 the effective potential vanishes: generic paths are scale-invariant spirals with constant radial speed, while the special case E = 0 gives neutrally stable circles. For η > 1 the motion is ‘fall to the centre’ in finite time: for E < 0 there is a single turning point (an apocentre) followed by plunge, whereas for E > 0 capture occurs from infinity without any turning point.