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(disclaimer: I am not a physicist).
I have been taught about the concept of energy in a fairly ad-hoc way. i.e. I was simply "told" that kinetic energy equals $\frac 1 2 m v^2$, and similarly I was told what potential energy is. Then I was shown how energy is conserved over time (in a classical system).
The concept of energy sort of intuitively makes sense to me, but I'd like a better understanding than just this ad-hoc one. At one point I thought of energy as "a basic quantity that is preserved over time in physical systems as the result of the underlying equations of motion". This seems true, but there are other quantities that are preserved, such as momentum, or the function $f(s)=5.6$ which maps every element of state space to $5.6$. Hence it doesn't really help me understand the concept.
Moreover, the concept of energy emerges in wildly different areas of physics (e.g. in classical, quantum, and general relativistic physics), where the concept seems to have the same general meaning despite the differences. I can even imagine that the concept of energy could apply to dynamical systems that have absolutely nothing to do with the physical world, such as certain completely artificial simulations, so long as they satisfy certain properties.
What is the most general conception/definition of energy? i.e. one that makes the least amount of assumptions about the underlying system (perhaps even so that the underlying system is not a "physical" system but an element from some more general set of dynamical systems).
I know that energy is conserved, but this doesn't distinguish it from other conserved quantities.
Noether's theorems imply that there is a quantity that is conserved due to time translation invariance. Is it conceptually insightful to see this as the "definition" of energy? If not, what is the most general definition of energy? And why does time translation invariance induce a conserved quantity that has such an important role in physics?