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

TL;DR

This paper explores how SE(d)-symmetry reduction helps understand steady swimming as a limit cycle, revealing stable fixed points and the potential for robust periodic orbits in fluid dynamics.

Contribution

It demonstrates that symmetry reduction simplifies the analysis of swimming, identifying stable fixed points and suggesting the existence of approximately periodic orbits in the reduced system.

Findings

Reduction by SE(d)-symmetry reveals a stable fixed point for a motionless body.

Speculation on the existence of approximately limit cycle orbits in the reduced phase space.

Lifting these orbits yields robust relatively periodic swimming motions.

Abstract

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.