When two billiard balls collide, the instantaneous forces between them are very large but...

When two billiard balls collide, the instantaneous forces between them are very large but act only in an infinitesimal time Δt, in such a manner that the quantity

remains finite. Such forces are described as impulsive forces, and the integral over Δt is known as the impulse of the force. Show that if impulsive forces are present Lagrange’s equations may be transformed into

where the subscripts i and f refer to the state of the system before and after the impulse, Sj is the impulse of the generalized impulsive force corresponding to qj, and L is the Lagrangian including all the nonimpulsive forces.