Instability of oscillations in an electrostatic potential

A popular exercise appearing in many undergraduate textbooks of introductory physics courses is re-examined. We analyze in detail the motion of a charged particle started from rest near the center of a charged ring. When the initial position of the particle is exactly along the central axis, simple harmonic motion is expected. With small deviations away from the axis, the oscillations sustain only a few cycles. We suggest that the instability of oscillations in electrostatic potentials can be demonstrated by this exercise.