 Solution

For fixed nonzero $l=p_{\varphi }$, the radial effective potential is $V_{\mathrm{eff}}(s)=l^2/(2m{s}^2{\sin }^2\theta )+mgs\cos \theta$. Its stationary point isThe equilibrium is therefore stable. A Taylor expansion with $x=s-s_0$ and $p_x=p_s$ givesQualification: the printed “any” $p_{\varphi }$ excludes a regular interior equilibrium when $p_{\varphi }=0$. In that case $V_{\mathrm{eff}}=mgs\cos \theta$ has no stationary point for $s>0$. Its minimum is the nonsmooth apex, where these coordinates and the smooth-cone dynamics fail.