Hamilton's principle.

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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.

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