On efficiency calculations for nonholonomic locomotion problems: An application to microswimming

We compare two notions of efficiency for locomotion processes modelled by l inear variational problems with nonholonomic constraints. We revisit Lighth ill's work on squirming motions of a sphere inside a viscous fluid, also st udied by Blake and by Shapere and Wilczek. Our approach uses the spectral b asis of the propulsion operator (sending velocity to surface force fields) to describe the surface deformations. We show that Shapere and Wilczek's ex tremal solutions, using high order surface deformations, are optimal only i n a naive sense. In fact, using Lighthill's efficiency definition, high ord er deformations are not efficient. Finally, we present a new criterion whic h combines the advantages of the two notions. MSC (1991): 76D07, 76Z10, 93B 29, 93C10, 51P05, 53C05.