There's an interesting conversation on Energy Conservation going on under my post about how The Universe can't `Just Exist'. I wrote a reply in that thread which I've decided to turn into a main post.
Saving Energy Conservation:
What would Physicists do if an Experiment showed that Energy Conservation is False?
Just because current experiments seem to show that energy is conserved, doesn't mean a future experiment might not violate energy conservation.
However, what this would actually look like in practice, would be a complicated dialectic involving both experiment and theory.
So suppose for example we do an experiment and it looks like under such-and-such conditions, a box of electrons seemed to get new kinetic energy seemingly coming out of nowhere. Then I think scientists would probably go through something like the following stages (not entirely unlike the silly pop-psych notion of "Stages of Grief")
1. The first thing physicists would do is suspect that there is some sort of experimental measurment error. In other words, reject the experimental result on the basis of well-established theory, and try to look for a reason why the mistake was made. Either because of a measurement error, or because of leakage from some established type of energy (like thermal leakage or external electric fields or something).
[This is like the "Denial" stage of grief, but unlike when somebody dies, most of the time when a scientific experiment discovers something really weird, denial is the right first reaction! Maybe there might be a bit of "Anger" too if the experimentalists are crackpots or making a slipshod mistake, and refuse to accept correction.]
2. If scientists conclude the effect is real, then they would try to find a new physical theory which explains the effect. Most conservatively, by adding a new type of particle or interaction. Importantly, we would normally try to redefine energy in such a way that it is still conserved in the new theory, even if we have to add a new term to the formula for energy.
[I guess this is the "Bargaining" stage of grief, but again if this is successful there is no need to go onto the next stages...]
3. If, after repeated efforts, it appears that there is NO reasonable or natural way to modify our physics theories (and associated definitions) in such a way that there is a conserved energy, then this would rise to the level of a crisis in physics ["Depression"], and people would start looking for more radical solutions to the problem.
4. But if it was discovered that there ARE simple ways to explain the data, by using theoretical models which DON'T have any interesting or useful concept of energy conservation; then (and only then) would we conclude that Energy Conservation is false. ["Acceptance"]
In other words, the first instinct of physicists would be to try to save the paradigm of the theory of Energy Conservation, and there are lots of different ways that this can be done--including questioning the experiment, postulating new physics, and subtle redefinitions of what we mean by "energy". This might lead one to think that physicists have an absolute a priori commitment to energy conservation. But I don't think that's true. I think the paradigm of Energy Conservation is like a piece of toffee, where you can stretch it a lot, but if you stretch it too much it will break. In other words, there exists the potential for future experiments to drive sufficiently radical changes of our theories, that the concept of energy is no longer applicable (except in whatever approximations are needed for the old paradigm to hold).
As I said before, one can certainly write down differential equations which do not satisfy a conservation law, for any quantity that resembles an energy. (A simple example would be if the laws of Nature turn out to be explicitly time dependent. In that case Noether's proof of energy conservation would not apply, and energy could be created or destroyed!