The principle conservation of energy is often taken as an obvious fact, or law of nature. But it seems to me the definition of energy is far from obvious, or natural: http://en.wikipedia.org/wiki/Energy lists lots of different types of energy.
So if I want to apply this principle in some concrete experiment, I have to go through all the forms of energy and consider whether this form of energy is applicable to each particular entity in my experiment. This seems like a rather cookbook-oriented approach (and the wiki list doesn't even claim to be complete!).
Now I wonder: ---> to what extent can these different energies be derived from some single simpler definition?
For example, if my model is that everything is made up of atoms (I don't want to consider anything at a smaller scale, I fear that would muddle the discussion and miss the main point. Also, I'm considering only classical mechanics.) which are determined by their position, momentum, charge and mass (?), is there a clear and exhaustive definition of the energy of a such a system?
EDIT: In light of comments and answers, I think I need to clarify my question a bit.
Is it true that the electric potential and gravitation potential (for atoms, say) will explain all instances of conservation of 'energy' occurring in classical mechanics?
If no, is there some modification of "electric potential and gravitation potential" above which will yield yes?
My question is not really about mathematics - Noether's theorem for example is a purely mathematical statement about mathematical objects. Of course mathematics and my question are related since they both involve similar kinds of reasoning, but I'm ultimately after a physical or intuitive explanation (which is not possible using only mathematics since this involves choice of a model, which needs to be explained intuitively) or assertion that all these energies (chemical, elastic, magenetic et.c. (possibly not including nuclear energy - let's assume we're in the times when we did not know about the inner workings of atoms)) come from some simple energy defined for atoms (for example).