This paper proposes a novel flexible coupler antenna (FCA) that translates passive coupling elements around a fixed-position active antenna to reshape the induced currents on the passive elements for radiation. A new form of mechanical beamforming is achieved by moving only the passive coupling elements while keeping the active antenna stationary. The proposed design significantly reduces the antenna and radio-frequency (RF) chain costs of conventional active array beamforming with low mechanical control complexity and energy consumption. For the purpose of exposition, we consider a point-to-point communication system with one FCA at the transmitter and one fixed antenna at the receiver. Specifically, based on multi-port circuit theory, we establish both the line-of-sight (LoS) and multipath channel models and derive the mechanical beamforming weights of the passive couplers as functions of their positions. Then, we formulate a new problem to maximize the received signal-to-noise ratio (SNR) by optimizing the positions of passive couplers at the transmitter, subject to coupler movement and transmit power constraints. Solving the resulting problem is inherently difficult because coupled channel and mechanical beamforming create non-linearity in the objective function.To tackle this problem, we propose an efficient block-coordinate conditional gradient method to search for the best positions of all passive couplers by sequentially optimizing the position of each coupler with those of the other couplers fixed in an iterative manner.Simulation results demonstrate that the proposed system significantly outperforms benchmark schemes in terms of achievable rate, but with significantly reduced active antennas and RF chains.