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Supppose we place a point charge inside a static electric field, with zero initial velocity.

I read in some notes, that its trajectory will be identical to the field line crossing that point. I can understand that this will be the case if the field lines are straight lines, for example in a central field (i.e. an electric field generated by some point charge), or an homogeneous field (inside a capacitor for instance), but I do not have a clear understanding or a concrete argument for the general case. For example, would that be the case for an inhomogeneous field like:

                                     enter image description here

I have seen this: Are the field lines the same as the trajectories of a particle with initial velocity zero, obviously relevant question, but i cannot say that i am fully convinced by the arguments in the answers. Also, i would hope for an answer based on the investigation of the differential equations of motion (given that we know the equations of the field lines) rather than on purely qualitative arguments.

Finally, let me mention, that i am mainly interested in the case of fields in vacuum (rather than inside some media).

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I'm not sure what you find unconvincing about those other answers, it is certainly just not true that the trajectory follows field lines in general. The reason we know this is that to follow any curved trajectory requires a component of force perpendicular to the trajectory, and of course the electric field line will only produce a force component along the field line.

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The field lines show the direction of the force on a charged particle, and therefore the acceleration of that charged particle.

If the particle is stationary then it will start moving in the direction of the applied force so initially it will start following the field line. However since the field line shows the direction of the acceleration not the direction of the velocity, once the particle it has a non-zero velocity it will diverge from the field line.

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