Optimal nonequilibrium engines obey the thermodynamic Lorentz force law

What are the fundamental limitations for finite-time work extraction from a non-conservative overdamped Langevin system, and what are the optimal protocols that achieve them? We show that the finite-time work extraction may be rewritten as a Lorentz-force Lagrangian action, with the kinetic term corresponding to a thermodynamic metric term that is an L2-optimal transport cost for the time-dependent probability density, and the magnetic field coupling term corresponding to an effective quasi-static work extraction. Optimal protocols are protocols that steer the thermodynamic state trajectory to satisfy a Lorentz force law defined on thermodynamic state space, through the inclusion of an additional counterdiabatic driving term. Classical results in electromagnetism such as gauge invariance, conserved quantities, etc. may then be ported over. Furthermore, our results prove mathematically that the recently-studied constant-velocity angle clamp protocol experimentally applied to the F1 molecular motor that suppressed nonequilibrium variation, is actually the globally optimal protocol.