We study current-induced domain wall dynamics in a thin ferromagnetic nanowire. We derive the effective equations of domain wall motion, which depend on the wire geometry and material parameters. We describe the procedure to determine these parameters by all-electric measurements of the time-dependent voltage induced by the domain wall motion. We provide an analytical expression for the time variation of this voltage. Furthermore, we show that the measurement of the proposed effects is within reach with current experimental techniques. We also show that a certain resonant time-dependent current moving a domain wall can significantly reduce the Joule heating in the wire, and thus it can lead to a novel proposal for the most energy efficient memory devices. We discuss how Gilbert damping, non-adiabatic spin transfer torque, and the presence of Dzyaloshinskii-Moriya interaction can effect this power optimization. Furthermore, we propose a new nanodot magnetic device. We derive a specific time-dependent current that is needed to switch the magnetization of the nanodot the most efficiently.