A Lagrangian Approach to Modeling and Analysis of a Crowd Dynamics

Modeling of crowd and pedestrian dynamics has intrigued engineers, physicists, and sociologists alike, even in recent times. Scientists have long sought to model the collective motion of large groups of individuals and study the mathematical basis of what seems to be apparently random behavior. A large number of macroscopic models have been proposed that describe crowd motion as a whole, much like the partial differential equations of fluid mechanics. This paper proposes a Lagrangian approach to the modeling of crowd dynamics by taking into consideration the various forces that act between the members of a crowd while they are in motion in a 2-D field. We attempt a realistic modeling of the attractive and repulsive forces between the members and seek to give a definite mathematical backbone to the terms "panic" and "evacuation." That the dynamics is stable is demonstrated by constructing an appropriate Lyapunov energy function. We then linearize the dynamics to obtain mathematical expressions for the small perturbations about an equilibrium point. Through machine simulations and by tracking the motion of actual crowd systems, we show the validity of the mathematics of group formation and evacuation that we have proposed.