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

We compare two notions of efficiency for locomotion processes modelled by linear variational problems with nonholonomic constraints. We revisit Lighthillʹs work on squirming motions of a sphere inside a viscous fluid, also studied by Blake and by Shapere and Wilczek. Our approach uses the spectral basis of the propulsion operator (sending velocity to surface force fields) to describe the surface deformations. We show that Shapere and Wilczekʹs extremal solutions, using high order surface deformations, are optimal only in a naive sense. In fact, using Lighthillʹs efficiency definition, high order deformations are not efficient. Finally, we present a new criterion which combines the advantages of the two notions.