Table 1 Models of symmetric swimmer encountered in the literature

∙ Scallop

\(\dot x_{0} = \frac {\dot \theta \,\sin \theta }{2-\cos ^{2}\theta }\)

∙ Symmetric 2-links Copepod

\(\dot x_{0} = \frac {\sum _{i=1}^{2} \dot \theta _{i} \sin \theta _{i}}{\sum _{i=1}^{2} \left (1+\sin ^{2}\theta _{i}\right)}\)

∙ Symmetric Takagi n-links equal lengths L=1

\(\dot x_{0} = \frac {(1+2r)\,\sum _{i=1}^{n} \dot \theta _{i} \sin \theta _{i}}{3/2\,r\log (2/r)\sum _{i=1}^{n} \left (1+\sin ^{2}\theta _{i}\right)}\)

∙ Symmetric Purcell (Avron-Raz),

\(\mathrm {d} x_{0} = \frac {\mathrm {d} \xi _{1}+\mathrm {d} \xi _{2}}{4+c-\xi _{1}^{2}-\xi _{2}^{2}}\)

2-links equal lengths L=1

\(\xi _{i} = \cos \theta _{i},\; c=l_{0}/2\)

Limit case: c=r=0⇒ Takagi = Purcell.