The Effective Gravitational Field of the Ball in Association Football

Physics > Physics and Society [Submitted on 1 Jul 2026] Title:The Effective Gravitational Field of the Ball in Association Football View PDF HTML (experimental)Abstract:In association football, collective player motion is organized around the ball. We ask whether this many-agent response can be described by an effective field analogous to gravitational attraction, while emphasizing that the measured quantity is a radial drift velocity of the players, as in an overdamped system, rather than a Newtonian acceleration. Using public tracking data from three anonymized games, we form two averages of the same player radial drift values: the distance-dependent radial drift velocity D(r), obtained by averaging at fixed player-ball distance, and the ball-position drift velocity field K(x,y), obtained by averaging the same inward velocity over observations in which the ball lies in a given pitch cell. After excluding restarts, other constrained phases of play, the nearest player to the ball, and separations r < 2 m, a persistent inward drift remains, with a global mean of 0.70 +/- 0.05 m/s. The distance dependence is not well described by inverse powers of r: moving outward from the ball, D(r) falls to a shallow minimum near r approximately 12 m, rises to a positive mid-range plateau, and then decreases toward zero at the largest separations. The full curve is summarized by a compact empirical drift law. The drift field K(x,y) is broadly positive over most of the pitch, but its three-game mean has a shallow central depression, about 25 percent below the non-goalmouth average, and stronger suppression near the central goalmouths, where the mean drift can become slightly negative. Including the player nearest to the ball or excluding goalkeepers leaves the radial law and drift-field pattern essentially unchanged. Football tracking data can therefore be used to extract simple effective laws of active many-body motion. Current browse context: physics.soc-ph Change to browse by: References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.