First-order periodic transverse swimming velocity
= First-order periodic transverse swimming velocity
{title2=$\langle V_1\rangle_t=\langle\Omega_1\rangle_t=0$}
The first-order lateral translation and rotation of a prescribed planar filament are time derivatives of fixed linear spatial averages of its shape. A periodic shape therefore gives zero temporal mean of both quantities. Net laboratory displacement can still arise at second order.