Optimal Rotations of Deformable Bodies and Orbits in Magnetic Fields

The optimal rotations of deformable bodies and orbits in magnetic fields were studied. The fundamental relation between the response (rotation) and the controls (deformation) was derived. It was found that deformations could induce rotation with zero angular momentum where dissipation was a natural cost function. It was also found that for certain plastic and viscous media in two different dimensions, the optimal path is the orbit of a charged particle on a surface of constant negative curvature with a magnetic field whose total flux is half a quantum unit.