Is there an easy (aka intuitive) way to understand that the conserved quantity from time translation symmetry is just what we call energy?
In other words, we use two definitions of energy. One is with Noethers theorem, and I've been told this is the fundamental one. The other is what is you learn in school and is mentioned in the examples below. The question is how to connect this two definitions.
- I can lift weights, so they get more energy.
- I can boil water for tee, so it gets more energy.
- I can burn $CaO$ with carbon, so I get carbide, that has more energy.
If I define energy as conserved quantity, how do I arrive at my examples..?
(Well, "easy/intuitive" is in the eye of the beholder. Thank you nevertheless.)
Side question: We have energy conservation in thermodynamics. I have never seen a Lagrangian formulation of thermodynamics. Can I only hope for an answer of my main question in theories that have such a formulation?