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This paper discusses the feasibility of beam tracking against dynamics in millimeter wave (mmWave) nodes placed on overhead messenger wires, including wind-forced perturbations and disturbances caused by impulsive forces to wires. Our main contribution is to answer whether or not historical positions and velocities of a mmWave node is useful to track directional beams given the complicated on-wire dynamics. To this end, we implement beam-tracking based on deep reinforcement learning (DRL) to learn the complicated relationships between the historical positions/velocities and appropriate beam steering angles. Our numerical evaluations yielded the following key insights: Against wind perturbations, an appropriate beam-tracking policy can be learned from the historical positions and velocities of a node. Meanwhile, against impulsive forces to the wire, the use of the position and velocity of the node is not necessarily sufficient owing to the rapid displacement of the node. To solve this, we propose to take advantage of the positional interaction on the wire by leveraging the positions/velocities of several points on the wire as state information in DRL. The results confirmed that this results in the avoidance of beam misalignment, which would not be possible by using only the position/velocity of the node.