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

Abstract : Based on copepod observations, Takagi proposed a model to interpret the swimming behavior 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 allow to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.
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Submitted on : Tuesday, April 25, 2017 - 8:27:34 AM
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Piernicola Bettiol, Bernard Bonnard, Alice Nolot, Jérémy Rouot. Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case. 2017. ⟨hal-01442880v2⟩

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