Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, copepod nauplii and copepod robot

Bernard Bonnard, Monique Chyba, Jéremy Rouot, Daisuke Takagi
2018 Pacific Journal of Mathematics for Industry  
The objective of this article is to present the seminal concepts and techniques of Sub-Riemannian geometry and Hamiltonian dynamics, complemented by adapted software to analyze the dynamics of the copepod micro-swimmer, where the model of swimming is the slender body approximation for Stokes flows in fluid dynamics. In this context, the copepod model is a simplification of the 3-link Purcell swimmer and is relevant to analyze more complex micro-swimmers. The mathematical model is validated by
more » ... l is validated by observations performed by Takagi's team of Hawaii laboratory, showing the agreement between the predicted and observed motions. Sub-Riemannian geometry is introduced, assuming that displacements are minimizing the expanded mechanical energy of the micro-swimmer. This allows to compare different strokes and different micro-swimmers and minimizing the expanded mechanical energy of the micro-swimmer. The objective is to maximize the efficiency of a stroke (the ratio between the displacement produced by a stroke and its length). Using the Maximum Principle in the framework of Sub-Riemannian geometry, this leads to analyze family of periodic controls producing strokes to determine the most efficient one. Graded normal forms introduced in Sub-Riemannian geometry to evaluate spheres with small radius is the technique used to evaluate the efficiency of different strokes with small amplitudes, and to determine the most efficient stroke using a numeric homotopy method versus standard direct computations based on Fourier analysis. Finally a copepod robot is presented whose aim is to validate the computations and very preliminary results are given.
doi:10.1186/s40736-018-0036-9 fatcat:sfmixg5pojdozbgx2ghkjkhrey