# Field lines and trajectories of zero initial velocity: are they identical?

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:

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).