Parking 3-sphere swimmer I. Energy minimizing strokes

Figure 1.  The swimmer sPr₃ is composed of three spheres of equal radii. The three spheres can move along three horizontal axes that mutually meet at $c$ with angle $2 \pi / 3$. The spheres do not rotate around their axes so that the shape of the swimmer is characterized by the three lengths $\xi_1, \xi_2, \xi_3$ of its arms, measured from the origin to the center of each ball. However, the swimmer may freely rotate around $c$ in the horizontal plane.

Figure 2.  An observer watching the dynamics $\gamma(p_0, \xi)$ of sPr₃ projected on a glass, imposes to another observer, lying on the other side of the glass, to watch the dynamics $\gamma (p_0, L \xi)$ of a micro-swimmer obtained from sPr₃ by inverting arms ${\left. \right\|_\boldsymbol{1}}$ and ${\left. \right\|_\boldsymbol{2}}$.

Figure 3.  (left) With respect to the standard euclidean space $\left( \mathbb{R}^3, (\cdot, \cdot)_2 \right)$, the energy minimizing strokes able to reach a prescribed displacement $\delta p$ are ellipses of $\mathbb{R}^3$ centered at the origin and contained in the plane spanned by the vectors $u : = U_{} a_1$ and $v : = U_{} b_1$. (right) With respect to the inner-product space $\left( \mathbb{R}^3, (\cdot, \cdot)_{\mathfrak{g}}\right)$, the energy minimizing strokes describe circles of radius $\sqrt{|\omega |}$.