Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case^★

¹ Laboratoire de Mathématiques Unité CNRS UMR 6205, Université de Bretagne Occidentale, 6, Avenue Victor Le Gorgeu, 29200 Brest, France.
² Institut de Mathématiques de Bourgogne et Inria Sophia Antipolis, 9 avenue Savary, 21078 Dijon, France.
³ Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France.
⁴ LAAS-CNRS, Université de Toulouse, LAAS, 31031 Toulouse Cedex 4, France.

^* Corresponding author: piernicola.bettiol@univ-brest.fr

Received: 24 April 2017
Accepted: 27 October 2017

Abstract

In Takagi [Phys. Rev. E 92 (2015) 023020], based on copepod observations, Takagi proposed a model to interpret the swimming behaviour of these microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and compares them invoking the concept of efficiency. Our aim is to provide an interpretation of Takagi’s results in the frame of optimal control theory and sub-Riemannian geometry. The maximum principle is used to select two types of periodic control candidates as minimizers: sinusoidal up to time reparameterization and the sequential paddling, interpreted as an abnormal stroke in sub-Riemannian geometry. Geometric analysis combined with numerical simulations are decisive tools to compute the optimal solutions, refining Takagi computations. A family of simple strokes with small amplitudes emanating from a center is characterized as an invariant of SR-geometry and allows to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.