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The given Lagrangian is: [ L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} - e\phi + e \vec{A} \cdot \vec{v} ] We are to find the equations of motion using the Euler–Lagrange equation: [ \frac{d}{dt} \left( \frac{\partial L}{\partial \vec{v}} \right) = \frac{\partial L}{\partial \vec{r}} ] Step 1: Compute (\frac{\partial L}{\partial \vec{v}}) [ \frac{\partial}{\partial \vec{v}} \Big[ -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \Big] = -mc^2 \cdot \frac{1}{2} \cdot (1 - \frac{v^2}{c^2})^{-1/2} ______ _____ _________ ___ ____ ______ ______ ____ ___ ____ _________ _______.
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