= Paul trap secular-motion example
{c}
{title2=$a_x=0,\ q_x=0.10,\ q_z=-0.20$}
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 are
$$
\frac{d^2x}{ds^2}=\frac{q_x}{2}\cos s\,x,\qquad
X=X_0\cos\left(\frac{q_xs}{2\sqrt2}\right),\qquad
x_{\rm app}=X\left(1-\frac{q_x}{2}\cos s\right).
$$
Set $x(0)=0.95X_0$ 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.
\Image[/paul-trap-rf-and-secular-motion.png]
{title=Full RF motion and the leading secular approximation}
{description=Original chosen small-parameter calculation in units of $X_0$. The lower panel resolves six drive cycles and includes the first micromotion correction. Agreement over this finite interval does not establish a stability diagram or an exact secular frequency.}
Using 400, 800, 1600 and 3200 steps per RF period gives successive maximum position differences of approximately $1.40\times10^{-4}$, $3.50\times10^{-5}$ and $8.74\times10^{-6}$ 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 $2.92\times10^{-6}X_0$, whereas $x_{\rm app}$ 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 $|q_i|$.
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