Force-free two-sphere stroke
= Force-free two-sphere stroke
{title2=$\Delta X=\oint K(\ell)\,d\ell=0$}
For spheres linked by a length $\ell(t)$, axial mobility and $F_1+F_2=0$ imply $\dot X=K(\ell)\dot\ell$ for their midpoint. The displacement over a periodic stroke is $\oint K(\ell)d\ell=0$. Unequal radii permit instantaneous midpoint motion but not net translation. This is an explicit one-shape-coordinate example of the <scallop theorem>.