[Undergraduate Level] - In this video I finally prove Noether's theorem in general. I show that when you have a symmetry, you have a conservation law! I also show why when you don't have a symmetry, you don't have a conservation law!
This remark in particular drew my attentation: 17:42 "Noether's theorem is kind of like the Euler-Lagrange equation. So the Euler-Lagrange equation can be derived by thinking about all tiny variations to our position q but here we're only considering a specific symmetry variation and instead of getting the full Euler-Lagrange equations of motion what we receive is a conservation law. This theorem is like the equation of motion for symmetry variations and I think the proof given in many textbooks sort of obscure this simple logic; they write it in this notation that doesn't make it obvious." With other treatments of Noether's theorem I always had this gut feeling that something was missing in the narrative. I believe that this is that missing link. Application of the Noether procedure is to evaluate a specific subset of the total variation space.
I want ask again, but before i want say tanks because you answer my second question. In physic we believe that universe is isotropic and homogeneous, its mean if we do experiment in another location and time the result must be the same, but neother's theorem tell reverse it, if we do experiment in another time dan space the result change. How you explain this contradiction?
Noether's theorem doesn't say that. The two principles you stated are equivalent to saying that changing locations or changing times won't affect the outcome of an experiment. These are statements of symmetry, so Noether's theorem simply says that if they are true, they imply conservation of momentum and energy, respectively. It doesn't contradict the cosmological principle or the idea of repeatability.
I really like the videos on Noether theorem but you could speed up a bit. Watching them at less than 2x speed is unbearable and even then I am tempted to set the up a bit faster.
