What is the physical meaning of the action in Lagrangian mechanics?

The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian.

I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical meaning of action.

Answer

The action has no immediate physical interpretation, but may be understood as the generating function for a canonical transformation; see e.g., http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation