Identifying conserved quantities using Noether's theorem

[Snydes]

I've been asked to find the conserved quantities of the following potentials: i) U(r) = U(x^2), ii) U(r) = U(x^2 + y^2) and iii) U(r) = U(x^2 + y^2 + z^2). For the first one, there is no time dependence or dependence on the y or z coordinate therefore energy is conserved and linear momentum in the y and z direction are conserved. I'm having trouble with the angular momentum. It would seem to me that since there is only a dependence on x, that the lagrangian would be invariant under rotations around the y and z-axis and thus angular momentum in those directions is conserved. Similar approach for the other two potentials. Can anyone give me any more depth or background on this, what would be a concrete way to approach these types of problems so I can be more confident in my answer

 

[Orodruin]

since there is only a dependence on x, that the lagrangian would be invariant under rotations around the y and z-axis and thus angular momentum in those directions is conserved.

You are halfway there, but draw the wrong conclusion. Which axis can you rotate around without changing x?

 

[Snydes]

You are halfway there, but draw the wrong conclusion. Which axis can you rotate around without changing x?

If I rotate around the x axis, the x coordinate should remain unchanged then should it not, therefore the angular momentum in the x direction is conserved?

For the second case, would this mean that we are solely dealing with the xy plane, thus a rotation around the z axis leaves the system invariant and angular momentum in the z direction is conserved?

 

[Orodruin]

Correct.

 

[Snydes]

Correct.

What about the third case? my first though would be that since U(r(vector))=U(r^2(scalar)) then it is independent of any rotational angle (theta, phi) then angular momentum is conserved for all space. Is this true?

 

[Orodruin]

What about the third case? my first though would be that since U(r(vector))=U(r^2(scalar)) then it is independent of any rotational angle (theta, phi) then angular momentum is conserved for all space. Is this true?

Angular momentum is conserved in all directions (rather than "all space"). A general rotation has three angles (see Euler angles).