 Solution

Write $w=a^2{\Omega }^2/C$, requiring $C\ne 0$ 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 helixTo analyse a small initial radial velocity, eliminate the cyclic velocities using the conserved quantities. The radial energy equation is ${\dot{R}}^2/2+U(R)=\mathcal{E}$, withHere $U^{\prime }(a)=0$ and $U^{\prime \prime }(a)=2{\Omega }^2+C^2/a^2>0$. The circular cylindrical radius is therefore stable. For $\eta =R-a$, the first-order equation is $\ddot{\eta }+{\omega }_r^2\eta =0$, soThe orbit acquires small radial oscillations. The other coordinates are modulated consistently with the same constants:These expressions retain the stated initial angular and axial velocities. The perturbation adds ${\delta }^2/2$ to the conserved energy. The expansion assumes $|\delta |/(a{\omega }_r)\ll 1$, which follows from the supplied smallness condition when $\Omega \ne 0$; it is not an exact finite-amplitude helix.