= Optimal cylinder for a reciprocal-square constraint
For $q,C>0$, minimize a cylinder's <volume> $V=\pi R^2H$ over $R,H>0$ subject to $q^2/R^2+\pi^2/H^2=C$. Set $u=q^2/R^2$ and $v=\pi^2/H^2$. Then $u+v=C$ and $V=\pi^2q^2/(u\sqrt v)$. Minimizing <volume> is equivalent to maximizing $u^2(C-u)$ on $0<u<C$. Its unique interior maximum is at $u=2C/3$, $v=C/3$, with divergence of <volume> at both endpoints. Thus the unique global optimum has
$$
\frac HR=\frac{\sqrt2\pi}{q},\qquad
V_{\min}=\frac{3\sqrt3\pi^2q^2}{2C^{3/2}}.
$$
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