I'm trying to derive the Lagrangian for a non-relativistic charged particle under the influence of a magnetic potential.

I'm assuming that $F=-\textrm{grad}(V)$ and so by the Lorentz force we have $-\textrm{grad}(V)=q v \times (\textrm{curl}(A)) $ and thus I would somehow like to "solve" for $V$ in order to plug it into the Lagrangian $L\equiv T-V$.

However, doing so, I arrive at $-\partial_j V=q\sum_iv_i\left(\partial_j A_i-\partial_i A_j\right)$ and I don't see an obvious way to get from here to $V=-\frac{q}{c}\sum_iA_i v_i$.


1 Answer 1


Hint to the question (v2): For a velocity-dependent force ${\bf F}$ (such as e.g. the Lorentz force), the relationship between force ${\bf F}$ and potential $U$ is

$$ {\bf F}~=~\frac{d}{dt} \frac{\partial U}{\partial {\bf v}} - \frac{\partial U}{\partial {\bf r}}. $$

See e.g. Goldstein, Classical Mechanics, Chapter 1. See also e.g. this and this Phys.SE posts.


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