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.