[gmueckl]

Conservation of energy is simply the invariance of a system to translation in time: the system shows the same time evolution from a well defined starting state regardless of whether that state exists at a time t or a time t+t' with an arbitrary t'. This is one application of Noether's theorem.

This is an example of a topic that unstructured self study would likely skip. But it is included in every properly structured course because this theorem puts a fundamental explanation to all conserved quantities that occur in classical mechanics.

[not-a-llm]

Noeter theorem and energy get complicated in General Relativity, because there is no global time translation symmetry

https://curtjaimungal.substack.com/p/what-is-energy-actually

Ignoring that, saying "energy must exist because of Noether and time-translation" doesn't really tell what energy actually is, not a very satisfactory response

[gmueckl]

On the contrary, this is best description of what conservation of energy means: all externally generated forces acting on the system do vary with time. Im other words, this is a closed system as defined in thermodynamics.

The nature of energy is really just a result of this property.

The difficulties with energy in GR are related to the fact that those equations alone allow for space-times that are truly bizarre. Some that are technically solutions to the field equations are clearly nonphysical (e.g. Goedel's solutions). There is also a priori no reason why e.g. expanding metrics should actually yield proper conservation of energy.

[Xmd5a]

> Why is this conserved? Because ∇ᵤ(Tᵘᵥξᵥ) = (∇ᵤ Tᵘᵥ)ξᵥ + Tᵘᵥ(∇ᵤ ξᵥ).

Interesting.

> This is an example of a topic that unstructured self study would likely skip.

I'm not studying physics – could you take a look at some unstructured reflexions I wrote a couple weeks ago?

https://news.ycombinator.com/item?id=48699125

[BrandoElFollito]

Noether theorem is one of the nicest ones to derive (together with Maxwell equations). There is such an elegance to go from the axions to the result.

I still have a trauma associated with this, though, because it popped out in one exam and I could see the pages in my notes where it was derived but they were quite blank.

[ndriscoll]

I suppose it depends on what you mean by unstructured (vs. the original claim of self-study). e.g. my method of self-study was generally to find course requirements for a degree program, find corresponding course descriptions and syllabi, find the relevant books, course notes, and/or video lectures and proceed through those.

e.g. I did formal study of basic physics in an engineering program, but Noether's theorem was never mentioned. I came to that through self-study, and can't really imagine how anyone would miss it unless "self-study" means "read random blogs and watch random youtube videos." I expect even browsing Stack Exchange and Wikipedia would expose you to most core topics fairly quickly.

[gmueckl]

You are following a structured course. I meant unstructured in the sense that topics are not chosen based on a properly designed curriculum or syllabus, but based on personal preferences alone.