Instability of a surface elastic line on an unnatural ramp

A line with bending resilience (Hooke’s Law constant $b$) but invariant length $l$ (elastic line) is confined to a cylindrical surface of radius $R$. A clamp at its starting point requires its initial direction always to lie along the base circle, but its end point and direction are not constrained. The trajectory in which the elastic line lies for its entire length along the circle is then an equilibrium path. Since the elastic line can straighten by swerving toward the generators of the cylinder, however, the circumferential trajectory is unstable if $l/R$ exceeds the critical value $\pi /{2^{3/2}}\left ( { \approx 1.1107} \right )$. To improve stability, we have constructed a ramp along the circle in the form of a potential energy trough (depth $\gamma$, width $\delta$), which, by means of surface forces, acts to attract the elastic line toward the circle. When the ramp strength is built up to a level $\gamma \delta {R^4}/b \approx 5.1743$, the stability of the circumferential trajectory is somewhat improved, lengths in excess of $l/R \approx 1.4178$ now being unstable. But if the ramp strength is increased above this critical value, even if only slightly, the circumferential trajectory for arbitrarily long elastic lines abruptly becomes stable.