Euler-Lagrange Equation

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Given a Lagrangian $\mathcal{L}(x,\dot{x})$ over variables $x$ and velocities $\dot{x}$, the respective equations of motion of this Lagrangian are of the form

$$\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{x}}=\frac{\partial \mathcal{L}}{\partial x}$$

for each variable $x$.

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References