How rigorous can conservation of energy be made?

Question

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).

Answer 1

You are asking about classical mechanics, and you should read a bit lower in the article you linked to:

Classical mechanics distinguishes between kinetic energy, which is determined by an object's movement through space, and potential energy, which is a function of the position of an object within a field, which may itself be related to the arrangement of other objects or particles.

Energy is quite simple in classical mechanics and that is where the law of conservation of energy was verified experimentally ( Noether's theorem). The rest of manifestations came long after the establishment of classical mechanics as a self consistent theory of motion and statics.

Before this the article state why there is something you call a "cookbook method".

Whenever physical scientists discover that a certain phenomenon appears to violate the law of energy conservation, new forms are typically added that account for the discrepancy.

and I would add: and bring into a unified framework not only energy as such but all observations.

Statistical mechanics was an extension of classical mechanics into the microcosm of particles and explained beautifully how thermodynamic quantities emerged from microscopic conservation of energy (and much more) .

Electrodynamics came into line with the mathematics of classical mechanics, as new fields other than gravity and could be rigorously set up in the same theoretical framework.

So the above should answer your question, yes, there is a simple expression of conservation of energy which sums kinetic and potential energies in a consistent way and has not been invalidated macroscopically.

But if one is talking of atoms and molecules the creation of quantum mechanics became necessary exactly because conservation of energy could not be sustained in the classical manner of counting beans. Probabilities and expectation values had to be introduced into quantum statistical mechanics to resolve problems with energy as in the black body radiation The stability of atoms with electrons captured in stable orbitals about the nucleus was impossible in classical mechanics and electrodynamics.

The foundations of classical mechanics were hit by nuclear energy and special relativity had to be invented in order to keep energy and momentum conservation to extend Noether's theorem ( which is an elegant encapsulation of classical mechanics consergation laws).

And then astronomical observations forced General Relativity on us where the principle of conservation of energy becomes meaningless globally, and has meaning only in our small flat location .

Each of the frameworks has a region of validity where conservation of energy can hold and is validated, and each framework blends smoothly and consistently with the neighboring ones , albeit some complicated mathematics may be involved in showing consistency.

After all physics is about observations and elegant theories that can fit observations, and this is what we have to date, cookbook or not.

Answer 2

All energy can ultimately be described as the kinetic and potential energy of fundamental particles. The trouble is that this description is rarely useful.

For example, chemical energy is the energy released when atoms combine to form molecules, and this is just the rearrangement of the electrons and nuclei in the atoms that make up those molecules. So in principle the release of chemical energy in a reaction is just the changes in the kinetic and potential energy of electrons and nuclei and could be described as some sort of many body interaction. Actually theoretical chemists have consumed many CPU-years on computers calculating reactions by using exactly this method. The problem is that while it can be done the calculations are time consuming.

The reason we partition energy up into different forms is for convenience. For example in the calculations above we should really consider the nuclei as interacting protons and neutrons, or maybe even interacting quarks. However nuclear degrees of freedom are (usually) not excited during chemical reactions so we can just ignore them, and this greatly simplifies the problem. The downside is, as you say, that you have to think carefully about what you do and don't include in any calculation.