The Effective Gravitational Field of the Ball in Association Football

Abstract

In association football, collective player motion is organized around the ball. We ask whether this many-agent motion can be described by an effective field analogous to gravitational attraction, emphasizing that the measured quantity is a radial drift velocity, as in overdamped dynamics, rather than a Newtonian acceleration. Using public tracking data from ten matches, we characterize this radial drift velocity through the distance-dependent mean D(r) and the ball-position-dependent field K(x,y). After excluding restarts and other constrained phases of play, both goalkeepers, the player nearest to the ball, and small player-ball separations, a persistent inward drift velocity remains, with a global mean of about 0.9 m/s. The distance dependence is not described by an inverse power of r: D(r) decreases at short range and then forms a broad plateau that persists to the largest measured separations. The mean K(x,y) is positive over most of the pitch, with a central depression about 15 percent below the non-goalmouth average and stronger suppression near the goalmouths, where the mean drift velocity becomes slightly negative, corresponding to weak effective repulsion. The role-resolved D(r) shows a pronounced defender minimum near 12 m and a pronounced attacker maximum near 60 m, whereas only defenders produce effective repulsion near the goals in K(x,y). These results show that football tracking data can reveal simple effective laws of active many-body motion while retaining clear signatures of player role and pitch geometry.