Hamilton's principle.
According to Hamilton's principle,“The path actually traversed in a conservative, holonomic dynamical system from t1 & t2 is one over which the integral of the lagrangian between limits t1 & t2 is stationary (i.e the time integral of the lagrangian is extremum.) Analytically it can be represented as ∫L dt = J = extremum Where J is the extremum value of the time integral of the lagrangian and is known as Hamilton's principle function for the path. Or d[ ∫L dt ]=0 Where 'd' is the variation symbol. This principle helps us to distinguish the actual path from the neighbouring paths.