Past exam of the options I course of the University of Oxford / 2007 / H / H1

Time independence conserves the energy $E=T+mga\cos \theta$. The two cyclic coordinates giveThese are respectively angular momentum along the symmetry axis and angular momentum along the fixed upward vertical; $n$ is the axial spin rate. For $u=\cos \theta$ and $-1<u<1$, eliminate $\dot{\varphi }$ using $\dot{\varphi }=(h-Cnu)/[A(1-u^2)]$. The energy becomesMultiplying and rearranging givesTo justify the double-root condition, use the effective potentialEliminating the cyclic coordinates by Routh reduction gives the remaining equation $A\ddot{\theta }=-dU(\cos \theta )/d\theta$. A constant nonvertical angle therefore has $U^{\prime }(u_0)=0$ and $E=U(u_0)$. Since $P(u)=2A(E-U(u))(1-u^2)$, both $P(u_0)$ and $P^{\prime }(u_0)$ vanish. This argument uses the second-order equation; merely differentiating the energy identity at a stationary angle would not establish the derivative condition.

Thus steady nonvertical precession gives a common root in $(-1,1)$. The nonvertical qualification is implicit in the denominator used by the source. A top spinning exactly upright or inverted is also possible but corresponds to an endpoint $u=\pm 1$, not an interior root; the unrestricted literal claim would need that exception.

The distinction is captured by the double-root criterion for steady precession of a symmetric top.