= Fixed-speed energy optimum for a planar flagellum
{title2=$\beta_* = B/(B+\sqrt{B^2+AB})$}
In <resistive-force theory> with $0<\gamma<1$, head drag $\delta\geq0$ and the ideal constant-inclination condition $\alpha^2=\beta$, put $r=1-\gamma$, $d=1+\delta$, $A=r(\delta^2-\gamma)$ and $B=\gamma d^2$. At prescribed swimming speed $U$, eliminating wave speed using <linear resistive-force propulsion of a travelling filament> gives
$$
\frac{E}{\kappa_NLU^2}=\frac{(d-r\beta)(\gamma d+\delta r\beta)}{r^2\beta(1-\beta)}.
$$
Its derivative has numerator $A\beta^2+2B\beta-B$. This numerator is strictly increasing on $(0,1)$, is negative at zero, and equals $(\delta+\gamma)^2$ at one. Thus its unique zero is the displayed global minimum. When $A=0$ the formula gives $\beta_*=1/2$; when $\delta=0$ it gives $1/(1+\sqrt\gamma)$. A planar sawtooth realizes constant inclination in the ideal model; smooth rounding and elastic costs change that idealization.
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