This paper studies energy-efficient wireless communication using a rotary-wing UAV to transmit/receive data from multiple ground nodes (GNs). The goal is to minimize the total energy consumption of the UAV, including propulsion and communication energy, while meeting the communication throughput requirements of each GN. The UAV's trajectory and communication time allocation are jointly optimized. The problem is non-convex and involves an infinite number of variables, making it challenging to solve directly. To address this, the paper first considers a simple fly-hover-communicate design, where the UAV visits optimized hovering locations and communicates with GNs while hovering. This reduces the problem to finding optimal hovering locations, durations, and visiting order, which is NP-hard. However, by leveraging the traveling salesman problem (TSP) and convex optimization, an efficient approximate solution is obtained. Next, the paper considers the general case where the UAV communicates while flying. A new path discretization method is introduced to transform the problem into a discretized form with a finite number of variables. This allows the use of successive convex approximation (SCA) to find a locally optimal solution. The paper derives an analytical propulsion power model for rotary-wing UAVs, which is significantly different from that of fixed-wing UAVs. The energy consumption model includes both propulsion and communication energy, with propulsion energy depending on speed and acceleration. The paper also formulates the energy minimization problem, which involves optimizing the UAV trajectory, communication time allocation, and mission completion time. Numerical results show that the proposed designs achieve significant performance gains over benchmark schemes, demonstrating the effectiveness of the energy-efficient communication approach for rotary-wing UAVs. The paper also discusses the challenges of UAV communication, including limited onboard energy and the need for efficient energy use to enhance communication performance and extend UAV endurance. The study highlights the importance of trajectory optimization and communication scheduling in achieving energy-efficient wireless communication with rotary-wing UAVs.This paper studies energy-efficient wireless communication using a rotary-wing UAV to transmit/receive data from multiple ground nodes (GNs). The goal is to minimize the total energy consumption of the UAV, including propulsion and communication energy, while meeting the communication throughput requirements of each GN. The UAV's trajectory and communication time allocation are jointly optimized. The problem is non-convex and involves an infinite number of variables, making it challenging to solve directly. To address this, the paper first considers a simple fly-hover-communicate design, where the UAV visits optimized hovering locations and communicates with GNs while hovering. This reduces the problem to finding optimal hovering locations, durations, and visiting order, which is NP-hard. However, by leveraging the traveling salesman problem (TSP) and convex optimization, an efficient approximate solution is obtained. Next, the paper considers the general case where the UAV communicates while flying. A new path discretization method is introduced to transform the problem into a discretized form with a finite number of variables. This allows the use of successive convex approximation (SCA) to find a locally optimal solution. The paper derives an analytical propulsion power model for rotary-wing UAVs, which is significantly different from that of fixed-wing UAVs. The energy consumption model includes both propulsion and communication energy, with propulsion energy depending on speed and acceleration. The paper also formulates the energy minimization problem, which involves optimizing the UAV trajectory, communication time allocation, and mission completion time. Numerical results show that the proposed designs achieve significant performance gains over benchmark schemes, demonstrating the effectiveness of the energy-efficient communication approach for rotary-wing UAVs. The paper also discusses the challenges of UAV communication, including limited onboard energy and the need for efficient energy use to enhance communication performance and extend UAV endurance. The study highlights the importance of trajectory optimization and communication scheduling in achieving energy-efficient wireless communication with rotary-wing UAVs.