= Nonvertical qualification in the steady-top double-root condition
{title2=$P(u_0)=P'(u_0)=0,\quad-1<u_0<1$}
After fixing axial and vertical <angular momenta>, the heavy symmetric top has <energy> $A\dot\theta^2/2+V_{\rm eff}(\cos\theta)$. A fixed angle strictly between zero and $\pi$ requires both $E=V_{\rm eff}(u_0)$ and $V_{\rm eff}'(u_0)=0$. Thus $P(u)=2A(1-u^2)(E-V_{\rm eff}(u))$ has a repeated interior root. A vertical axis belongs to the excluded endpoints: the upright stationary top has $P(u)=2Amga(1-u)^2(1+u)$, whose only repeated root is one. An assertion of an interior common root therefore needs the nonvertical hypothesis.
Back to article page
