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The Wikipedia page on the Lorentz force states the following:

$$\boldsymbol{F}=q\left[- \boldsymbol{\nabla}(\phi-\boldsymbol{v} \cdot \boldsymbol{A})-\frac{d\boldsymbol{A}}{dt}\right]$$

which can take the convenient Euler–Lagrange form $$\boldsymbol{F}=q \left[- \boldsymbol{\nabla}_x(\phi-\boldsymbol{\dot{x}}\cdot \boldsymbol{A})+\frac{d}{dt}\boldsymbol{\nabla}_\dot{x}(\phi-\boldsymbol{\dot{x}}\cdot\boldsymbol{A})\right]$$

How did the author get from the former to the latter and what is the rationale? It seems like the author just find a convenient way to get to the end result by simply replacing the terms.

dmckee --- ex-moderator kitten
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newbie125
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1 Answers1

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Since we know that $$\nabla_{\dot{x}}(\phi) = 0$$

and

$$\nabla_{\mathbf{\dot{x}}} \left(\mathbf{\dot{x} \cdot A}\right) = \mathbf{A} $$

thus

$$\frac{d}{dt}\nabla_{\mathbf{\dot{x}}}\left(\phi - \mathbf{\dot{x} \cdot A}\right) = \frac{d}{dt}\mathbf{A}$$

user18764
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