Let say we have charged conductor $M$ and we know its potential energy function $V_m(r)$ when $M$ is isolated from any charges. We also have charged conductor $N$ with potential energy function $V_n(r)$ when it is isolated.
Now we put objects $M$ and $N$ close together, the charges on their surfaces redistribute. I am interested in potential energy at every point in space, can I still add potential energy functions $(V_m + V_n)$ to find that?
I'd like to think that it is possible to add functions like that, but I can't find a way of proving it mathematically yet. Any ideas of how to show this?
No, you can't add potentials like this, specifically because the charge redistributes. For an example, let $M$ be a small sphere and $N$ be a large disk. Let $M$ have some charge. The potential distribution is the same as a point charge of that magnitude. If $N$ has no net charge we know the potential distribution from it-zero everywhere. If we bring $M$ near $N$ it will see $N$ as (almost) an infinite plane and create an image charge on the other side, perturbing the potential distribution considerably.