Abstract:

Fish appear to swim by periodically moving their fins. Additionally, if they momentarily stop or swim through a tangle of seaweed they appear to recover speed very quickly soon after. This mixture of stability and oscillatory behavior suggests that swimming is a limit cycle. I will provide theoretical evidence which suggests the answer is ``yes''. This will be done by defining a transitive Lie groupoid which can be used for fluid structure interaction. The base of this groupoid will be a set of embeddings of a body into space. We will then define the Lagrangian on the Lie algebroid, and add a time periodic force to the shape of the body as well as a viscous dissipation force due to the non-zero viscosity of the fluid. We will then perform a reduction by SE(3) to obtain a Lie groupoid where the base is the shape space of the swimmer. We will go over some arguments which suggest that a limit cycle could exists in this reduced system. Assuming a limit cycle does exist, we can use reconstruction formulas to obtain the motion of the fish. However, many analytical issues remain to be discussed.