= Solution
{c}
If $gf''(a)>\omega^2$, define
$$
\nu^2=\frac{gf''(a)-\omega^2}{1+f'(a)^2}>0.
$$
The <linearized equation> is $\ddot\epsilon+\nu^2\epsilon=0$, with bounded <simple harmonic motion>
$$
\epsilon(t)=\epsilon(0)\cos(\nu t)+\frac{\dot\epsilon(0)}\nu\sin(\nu t).
$$
Thus the strict <stable equilibrium> condition is
$$
\boxed{f''(a)>\frac{\omega^2}{g}.}
$$
This is also a sufficient condition for local nonlinear <Lyapunov stability>. Indeed the equation for the <bead on a uniformly rotating wire> has the <conserved quantity>
$$
\mathcal H=\frac12\left(1+f'(r)^2\right)\dot r^2+gf(r)-\frac12\omega^2r^2.
$$
Its <derivative> is $\dot r$ times the governing equation moved to one side, so it vanishes. At $a$, the last two terms have zero first <derivative> and positive second <derivative> $gf''(a)-\omega^2$. Their strict local minimum traps sufficiently small perturbations. The motion is bounded rather than asymptotically returning to rest.
Back to article page
