| Autor: |
Bernard Bonnard, Monique Chyba, Jéremy Rouot, Daisuke Takagi |
| Jazyk: |
angličtina |
| Rok vydání: |
2018 |
| Předmět: |
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| Zdroj: |
Pacific Journal of Mathematics for Industry, Vol 10, Iss 1, Pp 1-27 (2018) |
| Druh dokumentu: |
article |
| ISSN: |
2198-4115 |
| DOI: |
10.1186/s40736-018-0036-9 |
| Popis: |
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. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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