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Gauge variation of the charged-particle Lagrangian

Under

ϕ \mapsto ϕ - \partial_{t} f

and

A \mapsto A + \nabla f

, the charged-particle Lagrangian changes by

L^{'} - L = q (\partial_{t} f + \overset{r}{˙} \cdot \nabla f) = q \frac{df ( r ( t ) , t )}{d t} .

It therefore has unchanged equations of motion by total-time-derivative invariance.

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