Is there an easy (aka intuitive) way to understand that the conserved quantity f

Question

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.

Examples

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?

Explanation / Answer

Falko, if I understand your question right, then this would be the following: how is it that getting some of that particular conserved quantity (called "energy") transferred into a (mechanical, chemical etc.) system may have such dramatic consequences as a phase transition (boiling water) or chemical bond making and breaking.

Intuitively speaking: how can you realize that it is exactly the Noether-current associated with time translation invariance, the one quantity necessary to boil your water?

I would try to attack the problem by pointing out that it comes to your (arbitrary) partitioning of the system under consideration, and whether you apply the symmetry (i.e. conservation law) to the entire system or to an arbitrarily out singled part of it. Let us take as an example the boiling of water for tea during an airliner flight.

As long as you consider the entire kitchen in the airplane, with the heater and its batteries, it is time translation invariant, the "energy" Noether current is conserved and nothing happens (Here we considered the entire kitchen/plane as a system A). If you define the water in the boiler as a system B in its own, then you break time translation invariance for B once you want to get a pot of tea (and you switch the button at a given time, so the hamiltonian of B becomes time-dependent!) This transfers some of the Noether conserved current (energy) between the battery and your boiler and effectively boils your water.

Now let us consider as system C a second plane flying in line of sight of your plane A. Pilot of C can see plane A but he neither is aware that you have left your seat and have put up a cup of tea, nor he sees any flashes or auras surrounding A as a result of yours having done this. For the pilot of C, system A is time translation invariant and its energy is conserved. Noether's theorem is thus corroborating his own empirical observations.

In summary: your breaking or not breaking of the symmetries of a system (and of the transfer of associated conserved current) depends very much on how accurately you delineate or circumscribe (within a fictitious separating membrane) your system under consideration. This is very similar to the method in Thermodynamics.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.