Magnetoelastic actuated motion of fine ferromagnetic particles

Wireless magnetic actuation offers precise control over microscopic devices, yet full planar manipulation of rigid, tethered magnetic particles remains challenging. We introduce a minimal variational model: a permanently magnetized planar ellipse anchored by two linear springs. First, we derive exact geometric conditions under which the springs can be configured so that the ellipse rotates freely without elastic penalty -producing a continuous family of zero-energy equilibria in which the ellipse's centre traces a closed loop dictated solely by spring geometry. Next, we incorporate a uniform in-plane magnetic field and prove that the equilibrium magnetization aligns uniformly with the field. In the so-called full-controllability regime -when the spring rest lengths are long enough -rotating the external field directly prescribes the ellipse's orientation: the particle follows its zero-energy trajectory to maintain magnetic alignment, achieving a global energy minimum. For shorter springs, zero-energy configurations exist over a restricted orientation range; outside this range, the ellipse is pinned at the origin. Our results yield exact criteria for planar control in this simplest magnetoelastic setting, offering clear guidelines for the design of microscale actuators and metamaterials.