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Principle of Least Action

The path integral formulation is rooted in the principle of least action, where the particle takes the path that minimizes the action integral.
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The statement of the theorem

The classical path xcl(t)x_{cl}(t) that minimizes the action integral S[x(t)]S[x(t)] is determined by the Euler-Lagrange equations, derived from the variation δS=0\delta S = 0: \n∂L∂x−ddt(∂L∂x˙)=0\frac{\partial L}{\partial x} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) = 0