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Action Integral

The action integral, S=∫t1t2extL(x,x˙,t)dtS = \int_{t_1}^{t_2} ext{L}(x, \dot{x}, t) dt, represents the time integral of the Lagrangian, central to the path integral.
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The statement of the theorem

Define the Lagrangian L(x,x˙,t)L(x, \dot{x}, t). The Action Integral SS is the time integral of the Lagrangian along a path x(t)x(t): \nS[x(t)]=∫t1t2L(x(t),x˙(t),t)dtS[x(t)] = \int_{t_1}^{t_2} L(x(t), \dot{x}(t), t) dt
Source: Wikipedia