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

Abstract

Abstract 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 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.

Keywords

• Geometric optimal control
• Sub-Riemannian geometry
• Optimality conditions
• Direct and indirect numerical methods in optimal control
• Micro-swimmers and efficient strokes
• Closed geodesics in Riemannian geometry