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About action

The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

I.e.

$$S=\int L\mbox{ d}t=\iiiint\mathcal L\mbox{ d}x^4$$

This is very useful in the Lagrangian reformulation of Classical Mechanics, where one may apply the to obtain the equations of motion.

Also useful in Quantum Mechanics, Quantum Field Theory, General Relativity, String Theory, and so on, due to the Path Integral and Variational formulations.