Non-constant potential distribution I know that given a static charge distribution, the closed loop integral of the electric field is zero.
But what if the charges are moving, i.e, the potential at a point a changing. Is the line integral around a closed still zero?
If not, how do we deal with such electrodynamic situation (i.e. names of a some techniques which I read about)?
 A: It depends on what you are studying. I came up with two different situations.
Situation #1: If you want to describe the real-life situation, then the answer is no, because moving charges generate magnetic fields. The rigorous equation describing the circulation of electric field is called the Faraday's induction law, and is formulated as follows:
$$ \int \vec{E} \cdot \vec{dl} = - \frac{d}{dt} \int \vec{B} \vec{n} \cdot dS. $$
It assumes that you already know the space-time distribution of the magnetic field $\vec{B}$, but generally you don't. This distribution is generated by charges and electric field itself. The complete mathematical system is then a system of (partial) differential equations with respect to charge trajectories and electric & magnetic fields. Equations you need are:


*

*Maxwell equations in their differential form (4 equations, 2 scalar & 2 vector = 8 total equations) - look them up on wikipedia or something. It is worth noting that the Faraday's law (mentioned above) is just one of these equations rewritten in the integral form.

*The Lorentz force which charges feel is $\vec{F} = q \vec{E} + q [\vec{v} \times \vec{B}]$. These are 3n additional equations (given you have n charges).


There is also another subtlety: our system becomes relativistic (because Maxwell equations are relativistic in their nature), so you have to use the Special Relativity dynamics instead of the Newtonian one (or you can choose to neglect relativistic dynamics in some special cases).
As you can see, this is much more complex than simple electrostatic problems. You really have to do some math.
Situation #2: If it is electrostatics you are studying, it is possible to neglect all the magnetic effects and consider the adiabatic motion of charges. In this case you also neglect the non-vanishing circulation of the electromagnetic field.
It can be argued that this situation is just an approximate version of the "true" first situation. But in reality we already know that the first situation is also an approximate version of something else which includes General Relativity, Quantum Field Theory and so forth. The most fundamental case is currently unknown, so both these two situations have the right to exist. You just have to choose what you can neglect and what you can't.
