# Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers
## 4 Optimal Strokes of NG, PMPY and VE Swimmers  
### Euler-Lagrange Equation for Optimal Strokes of LRN Swimmers  
(Hereafter we will continue our discussion using the non-dimensional quantities but omitting the \(*\) notation for simplicity). As is introduced in Section 2, a swimming stroke \(\gamma(t)\) is a \(t\)-series of swimmer shapes. In the case when the swimmer has only finitely many degrees of freedom in their shape deformations, \(\gamma\) can be considered as a function from \([0,T]\) to \(\mathbb{R}^{m}\), where \(m\) is the number of degrees of freedom of the swimmer, and each component in \(\gamma(t)\) defines one degree of freedom. For example, all three linked-sphere swimmers in Sections 3.3-3.5 have only 2 degrees of freedom, thus we have \(\gamma_{\text{NG}}(t)=(l_{1}(t),l_{2}(t))^{T}\) for a NG swimmer, \(\gamma_{\text{PMPY}}(t)=(l(t),a_{1}(t))^{T}\) for a PMPY swimmer, and \(\gamma_{\text{VE}}(t)=(a_{1}(t),a_{3}(t))^{T}\) for a VE swimmer.  
The linear and driftless properties of Stokes flows make the LRN swimming problem a classic driftless controllable system [25; 28; 31; 32; 35; 43; 44]. Consider a cyclic stroke \(\gamma(t)\) of a LRN swimmer, where \(t\in[0,1],\ \gamma(0)=\gamma(1)\). The net translation \(X(\gamma)\) and energy dissipation \(\mathcal{E}(\gamma)\) within a stroke can be represented as:  
\[X(\gamma) = \int_{0}^{1}U(\gamma)dt=\int_{0}^{1}\mathcal{F}(\gamma)\cdot \dot{\gamma}dt \tag{10}\] \[\mathcal{E}(\gamma) = \int_{0}^{1}P(\gamma)dt=\int_{0}^{1}\big{(}\mathcal{G}(\gamma) \dot{\gamma},\dot{\gamma}\big{)}dt \tag{11}\]  
where for NG, PMPY and VE equations, the operators \(\mathcal{F}\) and \(\mathcal{G}\) can be calculated from the equations from Sections 3.3-3.5, respectively. For a given initial shape which we denote by \(\gamma_{0}\) and a given net translation \(X_{0}\), let \(\Gamma\{\gamma_{0},X_{0}\}\) be the set of all cyclic strokes that starts and ends with shape \(\gamma_{0}\) and results in a net translation of \(X_{0}\) in a cycle:  
\[\Gamma\{\gamma_{0},X_{0}\}=\{\gamma\big{|}\gamma(0)=\gamma(1)=\gamma_{0},X( \gamma)=X_{0}\}\]  
The optimal stroke problem can be formulated as follows:  
Given an initial shape \(\gamma_{0}\) and a net translation \(X_{0}\), find the stroke \(\overline{\gamma}\) in \(\Gamma\{\gamma_{0},X_{0}\}\) that minimizes the energy dissipation \(\mathcal{E}\):  
\[\mathcal{E}(\overline{\gamma})=\min_{\gamma\in\Gamma\{\gamma_{0},X_{0}\}} \mathcal{E}(\gamma) \tag{12}\]  
The Euler-Lagrange equation for this optimization problem is [22; 23]:  
\[-\frac{d}{dt}\big{(}\mathcal{G}\dot{\gamma}\big{)}+\frac{1}{2}\left(\begin{array} []{c}(\partial_{\gamma_{1}}\mathcal{G}\dot{\gamma},\dot{\gamma})\\ (\partial_{\gamma_{2}}\mathcal{G}\dot{\gamma},\dot{\gamma})\\ \vdots\\ (\partial_{\gamma_{m}}\mathcal{G}\dot{\gamma},\dot{\gamma})\end{array}\right)+ \lambda\big{[}(\
abla\mathcal{F})^{T}-\
abla\mathcal{F}\big{]}\cdot\dot{\gamma }=0 \tag{13}\]  
where \(m\) is the number of degrees of freedom in the swimmer's shape deformation, and \(\gamma=(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})^{T}\in\mathbb{R}^{m}\). Solutions of Equation (13) give the geodesics connecting \((\gamma_{0},0)\) and \((\gamma_{0},X_{0})\)which we refer to as the geodesic strokes. The optimal stroke is the geodesic stroke that consumes the least energy [22; 23].  
We use the _Shooting Method_ (SM) to numerically solve the governing Equation (13) for the optimal stroke problem. The numerical methods are developed in [22; 23] and have been applied for NG with equal sizes and PMPY swimmers; recent research works also report the efficiency of NG swimmers with unequal sized sphere [45]. In the following we use SM for all three (NG, PMPY and VE) linking-arm swimmers and compare their swimming performances. In Appendix A we present an outline of SM for the convenience of the readers of this paper.