Movable antennas (MAs) offer additional spatial degrees of freedom (DoFs) to enhance communication performance through local antenna movement. However, to achieve accurate and fast antenna movement, MA drivers entail non-negligible mechanical power consumption, rendering energy efficiency (EE) optimization more critical compared to conventional fixed-position antenna (FPA) systems. To address this issue, we develop a fundamental power consumption model for stepper motor-driven multi-MA systems based on electric motor theory. Based on this model, we formulate an EE maximization problem from a multi-MA base station (BS) to multiple single-FPA users. We aim to jointly optimize the MAs' positions, moving speeds, and the BS's transmit precoding matrix subject to collision-avoidance constraints during the multi-MA movements. However, this problem is difficult to solve. To tackle this challenge, we first reveal that the collision-avoidance constraints can always be relaxed without loss of optimality by properly renumbering the MA indices. For the resulting relaxed problem, we first consider a simplified single-user setup and uncover a hidden monotonicity of the EE performance with respect to the MAs' moving speeds. To solve the remaining optimization problem, we develop a two-layer optimization framework. In the inner layer, the Dinkelbach algorithm is employed to derive the optimal beamforming solution for any given MA positions. In the outer layer, a sequential update algorithm is proposed to iteratively refine the MA positions based on the optimal values obtained from the inner layer. Next, we proceed to the general multi-user case and propose an alternating optimization (AO) algorithm. Numerical results demonstrate that despite the additional mechanical power consumption, the proposed algorithms can outperform both conventional FPA systems and other existing EE maximization benchmarks