In this paper, we propose a prediction-based beam tracking scheme for the unmanned aerial vehicle (UAV) millimeter wave (mmWave) communication systems, where a UAV transmits to another UAV moving with relative high speed and rapidly changing angular velocity. Specifically, we first characterize the impacts of time-varying angular velocity and beam tracking frequency on the average transmission rate of our considered systems. Based on which, we obtain an analytical expression of the average transmission rate in the regime of high signal-to-noise ratio, thus enabling us to determine the optimal beam tracking frequency and predict the angle of arrival/angle of departure such that the maximum average transmission rate can be achieved. Through numerical results, we validate the effectiveness of our proposed adaptive predication-based beam tracking scheme, demonstrating how it can be robust to the rapidly changing angular velocity of the UAV, compared to various baselines.