= Solution
Write $w=a^2\Omega^2/C$, requiring $C\ne0$ and $a>0$. The initial conditions fix $\ell=a^2\Omega$ and $\dot z=w+C\log(R/a)$. At $R=a$, the radial force is $a\Omega^2-Cw/a=0$. Thus the unperturbed motion is the helix
$$
\boxed{R=a,\qquad\phi=\Omega t,\qquad z=wt.}
$$
To analyse a small initial radial velocity, eliminate the cyclic velocities using the conserved quantities. The radial energy equation is $\dot R^2/2+U(R)=\mathcal E$, with
$$
U(R)=\frac{a^4\Omega^2}{2R^2}+\frac12\left[w+C\log(R/a)\right]^2.
$$
Here $U'(a)=0$ and $U''(a)=2\Omega^2+C^2/a^2>0$. The circular cylindrical radius is therefore stable. For $\eta=R-a$, the first-order equation is $\ddot\eta+\omega_r^2\eta=0$, so
$$
\boxed{\omega_r^2=2\Omega^2+\frac{C^2}{a^2},\qquad
R=a+\frac{\delta}{\omega_r}\sin\omega_rt+O(\delta^2).}
$$
The orbit acquires small radial oscillations. The other coordinates are modulated consistently with the same constants:
$$
\phi=\Omega t-\frac{2\Omega\delta}{a\omega_r^2}(1-\cos\omega_rt)+O(\delta^2),\qquad
z=wt+\frac{C\delta}{a\omega_r^2}(1-\cos\omega_rt)+O(\delta^2).
$$
These expressions retain the stated initial angular and axial velocities. The perturbation adds $\delta^2/2$ to the conserved energy. The expansion assumes $|\delta|/(a\omega_r)\ll1$, which follows from the supplied smallness condition when $\Omega\ne0$; it is not an exact finite-amplitude helix.
Back to article page
