Also the Dirac delta function has following properties

                                                                                      (12.11)

In equation (12.9) by using the Direc delta function, it enables to represent the point force (concentrated) as the distributed load description. According to the Hamilton's principle for a dynamical system, the integral of Lagrangian function L = ( T+U-W ) taken over a given interval of time is stationary i.e.

                                                                                           (12.12)

where is a variational operator.                                                                                       

On substituting equations (12.5), (12.7) and (12.9) into equation (12.11), we get

                          (12.13)