  In this paper, we show how the motion of physical fields, in particular the
electromagnetic potential, is connected with the choice of a space and time
decomposition of the background spacetime manifold. The relation of the field
dynamics and its kinematic description is derived with the help of the
parametrization approach. In this context, we give an original proof for the
main statement of this approach. With regard to generally covariant theories,
the arising kinematic constraints are investigated for the class of non-linear
electrodynamics. We reduce the problem of finding the explicit form of the
super-Hamiltonian constraint to the problem of solving two non-linear
equations. By this, we provide a plug-and-play approach to find the whole set
of constraints for an arbitrary theory of non-linear electrodynamics and apply
it to selected examples. Furthermore, it will be shown that the parametrization
approach is connected with the description of a charged point particle within
its own electromagnetic field.
