I liked the article and I hope that I can understand it more with some study.

I think the following sentence in the article is wrong "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be horizontal, vertical, and angular momentum.”

I think the correct statement is "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be translation, rotation, and time shifting.”

I think translation leads to conservation of momentum, rotation leads to conservation of angular momentum, and time shifting leads to conservation of energy (potential+kinetic). It's been a few decades since I saw the proof, so I might be wrong.