Vector field theory in motion: Revealing latent potentials in soccer dynamics

Team sports offer a natural laboratory for studying collective motion in competitive environments. While recent tracking technologies enabled detailed analysis of player and ball trajectories, most existing approaches rely on discrete events and local kinematic descriptors. This limits the possibility of studying team collective behavior. Here we introduce a physics-based framework to characterize collective dynamics using empirically reconstructed velocity vector fields defined over the pitch. Taking soccer as a paradigmatic example, we show that both ball and player movements generate smooth, predominantly irrotational vector fields. Each of these fields is generated by a potential providing compact, physically interpretable representations of collective motion and encoding how teams control, or constrain play across space during attacking and defensive phases in each match. Applied to an entire season of top-level competition, the resulting potentials reveal stable team-specific fingerprints and quantitative associations between potential gradients and effective ball progression.