The role of SE(d)-reduction for swimming in Stokes and Navier-Stokes fluids

Steady swimming appears both periodic and stable. These characteristics are the very definition of limit cycles, and so we ask "Can we view swimming as a limit cycle?" In this paper we will not be able to answer this question in full. However, we shall find that reduction by SE(d)-symmetry brings us closer. Upon performing reduction by symmetry, we will find a stable fixed point which corresponds to a motionless body in stagnant water. We will then speculate on the existence of periodic orbits which are "approximately" limit cycles in the reduced system. When we lift these periodic orbits from the reduced phase space, we obtain dynamically robust relatively periodic orbits wherein each period is related to the previous by an SE(d)-phase. Clearly, an SE(d) phase consisting of nonzero translation and identity rotation means directional swimming, while non-trivial rotations correspond to turning with a constant turning radius.