On the role of reduction by symmetry in understanding swimming at mid-Reynolds

A number of numerical and experimental studies suggest suggest that swimming can be characterized as an emergent phenomena arising from time-periodic internal body forces. In particular, it seems reasonable to surmise that swimming can be characterized as a relative limit cycle. A relative limit cycle is a system trajectory with a time-period, wherein each period is related to the previous by the action of a Lie group. In the case of swimming in $R^n$ this Lie group is the set of rotations and translations, ${\rm SE}(n)$. In this talk we will describe a class of dissipative systems which admit relative limit cycles. Unfortunately, the Navier-Stokes equations coupled to a solids in $R^n$ are not within this class of. However, a Navier-Stokes-$\alpha$ fluid on the $n$-sphere, $S^{n}$, could resolve this issue. The relative limit cycles would be with respect to the group ${\rm SO}(n)$. In a very precise sense, the group ${\rm SO}(n)$ is to the $S^n$ as ${\rm SE}(n)$ is to $R^{n}$. As a result, the relative limit cycles obtained on $S^n$, can be characterized as spatially localized manifestations of trajectories for systems in $R^n$ wherein each period related to the next by a rigid rotation and translation.