# It is experimentally known that the equation of motion for a charge e moving in a static electric field E is given by: d/dt (gamma m v)=eE

It is experimentally known that the equation of motion for a charge $e$ moving in a static electric field $\mathbf{E}$ is given by:
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\gamma m\mathbf{v}\right)=e\mathbf{E}$
Is it possible to show this using just Newton's laws of motion for the proper frame of $e$, symmetry arguments, the Lorentz transformations and other additional principles?
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lydalaszq
If for some reason you'd like an explicitly relativistic formulation, take a look at the Lorentz force law:
$\frac{\mathrm{d}{p}^{\mu }}{\mathrm{d}\tau }=e{v}_{\nu }{F}^{\mu \nu }$
For the derivative with respect to coordinate time that you want, we need to multiply through by $d\tau /dt=1/\gamma$. But for a constant electric field in Cartesian coordinates, the only nonzero components of ${F}^{\mu \nu }$ are ${F}^{0a}=-{F}^{a0}$ for $a=1,2,3,$ which are the electric field components. Thus, only the ${v}_{0}=\gamma$ term can contribute, canceling factor brought in by time dilation factor. QED.
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Cierra Castillo
I think this is a lot simpler than you suspect. It's really just Newton's 2nd law, and recognising the concept of momentum.
$\frac{d}{dt}\left(\gamma mv\right)=\frac{dp}{dt}=F=eE$
Since classically the electric field $E$ is defined as the force $F$ divided by the elementary charge $e$.
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