Marginal straight-line launch at thirty degrees
= Marginal straight-line launch at thirty degrees
{title2=$\Delta\Phi_{\mathrm{eff}}/(GM/R_0)=-7(s/R_0)^3/16+O(s^4)$}
At exactly thirty degrees the quadratic curvature vanishes, but the specified straight-line point-mass <effective potential> is downhill for small positive displacement at cubic order. Thus the strict negative-curvature criterion does not exclude every nonlinear one-sided marginal launch. An exactly stationary bead remains at equilibrium until displaced.