Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, copepod nauplii and copepod robot
Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, copepod nauplii and copepod robot
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DOI：
10.1186/s40736-018-0036-9
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发表时间：
2018-06-22
影响因子：
--
通讯作者：
Takagi, Daisuke
中科院分区：
文献类型：
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作者：
Bonnard, Bernard;Chyba, Monique;Takagi, Daisuke
本文的目的是介绍次黎曼几何学和哈密顿动力学的开创性概念和技术，并辅之以适用的软件来分析桡足类微型游泳者的动力学，其中游泳模型是流体力学中Stokes流的细长身体近似。在这种情况下，桡足类动物模型是三连杆Purcell游泳者的简化，并与分析更复杂的微型游泳者相关。夏威夷实验室的高木团队进行的观测验证了该数学模型，表明预测的运动与观测到的运动之间是一致的。引入了次黎曼几何，假设位移使微泳者的扩展机械能最小。这允许比较不同的泳姿和不同的微泳者，并最小化微泳者扩展的机械能。其目标是最大化冲程的效率(冲程产生的位移与其长度之间的比率)。在次黎曼几何的框架下，利用极大值原理，分析产生笔划的一族周期控件，以确定最有效的一个。在次黎曼几何中引入的梯度范式用于评估小半径球体的效率，该技术用于评估小幅度不同笔划的效率，并使用数值同伦方法与基于傅立叶分析的标准直接计算相比较来确定最有效的笔划。最后给出了一个以验证计算为目的的桡足机器人，并给出了非常初步的结果。
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.