What is Energy?

This was a question posed on reddit by nexuapex.  The resulting answer by BoxAMu is very enlightening, so I have posted it here in its entirety.  

"To answer your question, first an interesting bit of history- In the 19th century, energy, or at least heat, was thought to be a physical substance. One of the great paradigm shifts in physics was the discovery that heat is just a form of motion. The misunderstanding with energy exists today because many textbooks and physicists still like to talk about energy as if it were a substance. Energy, from classical through quantum mechanics (I exclude general relativity since there it gets tricky and I am not an expert), is nothing more than a number. The only significance of it is that this number doesn't change. It's analogous to money in this way. We can't compare (for example) the value of an apple and an orange directly, but we do by assigning a dollar value to each. In the same way we use energy to compare different physical processes. An object in a gravitational field being set in to motion, for example. We use energy to define how much action of gravity this motion is 'worth'. It's said that potential energy is 'stored' energy, but that's completely misleading- in fact potential energy has no physical meaning at all. It's just a method of book keeping. The fact of gravity being conservative just means the book keeping is easy. If we know the displacement of an object in a gravitational field, we know how it's velocity will change. Compare to a non-conservative force, such as air resistance. In this case, the force is non-conservative because the energy of motion of the object being resisted is transferred to many air molecules. If we actually knew the velocities (and masses) of those air molecules, then in such a case air resistance would be conservative: we'd know the change in velocity of the object from the change in velocity of the molecules. So again the difference is only one of book keeping." -BoxAMu

"What are the conditions under which the actual "energy" number doesn't change? I know, for instance, that if you change reference frames, then your calculated energy changes. Are there more conditions?

Why is this "book-keeping" necessary? What math wouldn't work out if we didn't have potential energy around? Is a boulder rolling down a hill explainable without gravitational potential?" -nexuapex

"As other have pointed out, only changes in energy matter, not the absolute number. It's true that on top of this, even the changes of energy change in a different reference frame, but think about how this applies to doing an experiment. Take the classic example of throwing a ball back and forth on a train. One could calculate the motion of the ball and it's energy in the train frame or the ground frame. The actual numbers would be different in each case, but this does not prevent either observer from applying the laws of physics in their respective frame and making correct predictions. I believe the only condition is the usual one of physics- that the experiment or calculations are carried out in an inertial reference frame.

It's not that the book-keeping is necessary, it's just that it's really useful that we can even do it. The math of course does work out without potential energy- you can calculate the whole trajectory of a particle in the gravity example using the gravitational force, which is considered the more fundamental idea in classical mechanics. However, this type of reasoning gets more complicated beyond these basic classical mechanics calculations. Due to relativity (among other things), energy has been promoted to the more fundamental idea than force. Many modern theories are based on the Lagrangian formalism, which originally required the ideas of kinetic and potential energy. Now it's totally different, there's no basic force to derive a potential from- people just try come up with a Lagrangian that gives equations which make correct predictions (sorry field theory people if I'm oversimplifying). But energy again pops up as a conserved quantity, and is useful since it may simplify calculations." -BoxAMu