In unmanned aerial vehicle (UAV)-assisted wireless networks, the continuous trajectory designs generally suffer from infinite variable number and inherent trajectory continuity. In this paper, we aim at addressing the trajectory continuity and achieving globally optimal joint continuous trajectory and resource allocation design. We consider a UAV-assisted wireless network serving a ground deployed user and focus on the energy-constrained communication. Taking the practical power control capability into account, we target at maximizing the overall throughput via joint power control and UAV trajectory design. However, the practically discrete choices for power control have lead the problem highly intractable. We start with the ideal power control case and via dual analysis characterize the optimal power control as function of a dual variable and UAV position, such that the dual function can be transformed to pure trajectory design problems. Inspired by a novel mechanical equivalence idea, we convert pure trajectory designs into a series of equivalent mechanical rope equilibrium problems with adaptively defined potential field, which can be solved optimally via physical principles. Subsequently, an efficient algorithm is proposed for obtaining the optimal dual solution, from which the optimal joint solution is recovered in a closed form. Afterwards, the proposed design methodology is extended into scenario with practical discrete power control and corresponding optimal joint design is eventually achieved. Finally, compared with benchmark, the optimality and low design complexity in our obtained closed-form solution are verified via numerical results. The necessity of considering practical discrete power control is also highlighted.