Mathieu equation stability diagram (a,q) and animated ion trajectory
The Paul trap confines ions with oscillating electric quadrupole fields. The ion motion obeys the Mathieu equation: d²u/dτ² + (a − 2q cos2τ)u = 0, where a = 4eU/(mΩ²r₀²) and q = 2eV/(mΩ²r₀²). Stable trapping (bounded trajectories) occurs within the first stability region (shaded). The secular (slow) oscillation frequency is ω_sec ≈ (Ω/2)√(a + q²/2). Wolfgang Paul received the Nobel Prize in 1989.