This paper addresses the energy minimization problem for wireless communication using a rotary-wing unmanned aerial vehicle (UAV). The goal is to minimize the total UAV energy consumption, including propulsion and communication energy, while ensuring the communication throughput requirements of multiple ground nodes (GNs). The authors derive an analytical propulsion power consumption model for rotary-wing UAVs and formulate the energy minimization problem by jointly optimizing the UAV trajectory, communication time allocation, and mission completion time. The problem is non-convex and involves infinitely many variables over time, making it challenging to solve optimally. To tackle this, the authors first consider a simple *fly-hover-communicate* design, where the UAV successively visits hovering locations and communicates with GNs at each location. An efficient algorithm is proposed to optimize the hovering locations, durations, and flying trajectory using the traveling salesman problem (TSP) and convex optimization techniques. For the general case where the UAV communicates while flying, a novel *path discretization* method is introduced to transform the problem into a more tractable form with a finite number of variables. The successive convex approximation (SCA) technique is then applied to find a locally optimal solution. Numerical results demonstrate significant performance improvements over benchmark schemes in achieving energy-efficient communication with rotary-wing UAVs.This paper addresses the energy minimization problem for wireless communication using a rotary-wing unmanned aerial vehicle (UAV). The goal is to minimize the total UAV energy consumption, including propulsion and communication energy, while ensuring the communication throughput requirements of multiple ground nodes (GNs). The authors derive an analytical propulsion power consumption model for rotary-wing UAVs and formulate the energy minimization problem by jointly optimizing the UAV trajectory, communication time allocation, and mission completion time. The problem is non-convex and involves infinitely many variables over time, making it challenging to solve optimally. To tackle this, the authors first consider a simple *fly-hover-communicate* design, where the UAV successively visits hovering locations and communicates with GNs at each location. An efficient algorithm is proposed to optimize the hovering locations, durations, and flying trajectory using the traveling salesman problem (TSP) and convex optimization techniques. For the general case where the UAV communicates while flying, a novel *path discretization* method is introduced to transform the problem into a more tractable form with a finite number of variables. The successive convex approximation (SCA) technique is then applied to find a locally optimal solution. Numerical results demonstrate significant performance improvements over benchmark schemes in achieving energy-efficient communication with rotary-wing UAVs.