 Paul trap secular-motion example

Choose these dimensionless parameters for an original example, with no collisions or damping. They are teaching inputs, not measurements from either lecture. With $s=\Omega t$ and slow initial amplitude $X_0$, the radial equation and its leading approximation areSet $x(0)=0.95X_0$ and $dx/ds(0)=0$ to match the approximate initial conditions. The leading secular angular frequency is ${\omega }_x/\Omega =0.0353553$. The velocity Verlet method integrates the full RF equation over 80 drive periods; a separate fourth-order Runge-Kutta method checks the computed trajectory. Halving the timestep tests numerical error independently of the averaging error.

Using 400, 800, 1600 and 3200 steps per RF period gives successive maximum position differences of approximately $1.40\times 10^{-4}$, $3.50\times 10^{-5}$ and $8.74\times 10^{-6}$ in units of $X_0$, consistent with second-order convergence. The finest trajectory differs from the independent classical fourth-order Runge-Kutta method result by at most $2.92\times 10^{-6}{X}_0$, whereas $x_{\mathrm{app}}$ differs by as much as $0.0378X_0$. Over a sufficiently long time, the omitted correction to the secular angular frequency produces a growing phase discrepancy even if the numerical trajectory has converged. For the chosen point only, integration of a fundamental solution matrix over one RF period checks that both the radial and axial Floquet multipliers lie on the unit circle, away from its endpoints. This tests simultaneous ideal confinement without extrapolating to larger $|q_i|$.