If you go to the relativistic Hamiltonian, what allows us to go from(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\dot{\vec{P}} = - \frac{\partial \mathcal{H}}{\partial \vec{x}} = e (\vec{\nabla} \vec{A}) \cdot \dot{\vec{x}} - e \vec{\nabla} \phi [/tex]

to

[tex]\frac{d}{d t}\left(\frac{m \dot{\vec{x}}} {\sqrt {1 - \frac{\dot{\vec{x}}^2}{c^2}}}\right) = e \vec{E} + e \dot{\vec{x}} \times \vec{B}[/tex]

Is there some identity with

[tex]\frac{\partial A^i}{\partial x^i}[/tex]

that turns it into a curl that I'm missing?

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# Charged particle in EM field

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