Tracking objects in videos with non-linear motions is a challenging task. Unlike traditional multiple object tracking methods that often rely on linear motion assumptions and struggle to handle non-linear movements, this paper proposes a novel multiple object tracking method based on Test-Time Training (TTT) and Adaptive Iterative Scale-Up ExpansionIoU (AISE), namely TTT-MOT. In particular, the Test-Time Training is used to predict the non-linear motions of new frames based on current trajectories. And the Adaptive Iterative Scale-Up ExpansionIoU module with the deep ReID features are used for the association of detections and trajectories. Extensive experimental results demonstrate the effectiveness of our proposed method in tracking non-linear motion objects, achieving a score of 78.8% HOTA on the SportsMOT and 87.6% HOTA on the SoccerNet-Tracking dataset. It outperforms all previous state-of-the-art trackers, covering a wide variety of sports scenarios.