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.
I think your last paragraph is correct, but the statement in the article is referring to the specific 2D 2-body example given, and its original phrasing is also correct. Translation, rotation, and time-shifting are transformations (matrices), not quantities. Horizontal, vertical, and angular (2D) momentum are scalars. The article is saying that if you take the action potential given in the example, there exist scalar quantities (which we call horizontal momentum, vertical momentum, and angular momentum) that remain constant regardless of any horizontal, vertical, or rotational transformation of the coordinate system used to measure the 2-body problem.
In that sentence I was only talking about the translations and rotations of the plane as a group of invariances for the action of the two-body problem. This group is generated by one-parameter subgroups producing vertical translation, horizontal translation, and rotation about a particular point. Those are the "three degrees of freedom" I was counting.
You're right about the correspondence from symmetries to conservation laws in general.
The application of Noether's theorem in this case refers only to the energy integral shown (KE = ME - GPE for 2D Kinetic Mechanical and Gravitational Potential Energies) over time. It's really only for that particular 2 body 2 dimensional problem.
More generically in 3 dimensions a transformation with 3 translational 2 rotational and 1 time independence would provide conservation of 3 momenta 2 angular momenta and 1 energy.
Right, the rephrasing of the sentence is a tad more accurate. Your three entities are [invariant -> conserved quantity]: (translation -> momentum), (rotation -> angular momentum) and (time -> energy).
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.