Determine if angular momentum is conserved given potential energy funtions?

[ENT]

Homework Statement

For each of the following potential energy functions in three dimensions, what quantities are conserved (energy, momentum, angular momentum)?

Homework Equations

U = k/2(x^2 + y^2)

U = α/r

U = β(z(hat) dotted with r)^2

U = α/r + β(z(hat) dotted with r)^2

Where z(hat) is the unit vector in the z direction.

The Attempt at a Solution

So I understand that in order for the angular momentum to be conserved the cross product of r and F, being the force obtained from the negative gradient of U, must be equal to zero. However I am not sure what to use for the position vector r.

 

[haruspex]

Suppose you measure moment about a point v for a force F acting at r. You know Fxr=0. What will the moment about v be?

 

[ENT]

We don't know that F x r = 0. We are trying to determine whether it is or not, and I am not sure what to use for r.

 

[haruspex]

We don't know that F x r = 0. We are trying to determine whether it is or not, and I am not sure what to use for r.

Yes, sorry, I was thinking specifically of the U = a/r case, but forgot to say so.
As I read that question, U is specified in relation to a coordinate system where r is the position vector, and the field it defines will be symmetric about the origin. Hence Fxr will be 0.
For any other reference point v, what is the vector from v to the point r? If the force of the field at r is F(r), what will be its moment about v?

 

[paranormal]

To determine if angular momentum is conserved, we need to look at the potential energy functions and see if they have any dependence on angular position or angular velocity. If they do, then angular momentum would not be conserved.

Let's look at each potential energy function individually:

1. U = k/2(x^2 + y^2)

This potential energy function does not have any dependence on angular position or angular velocity. Therefore, angular momentum would be conserved.

2. U = α/r

This potential energy function does have a dependence on angular position, as it is inversely proportional to the distance r from the origin. Therefore, angular momentum would not be conserved.

3. U = β(z(hat) dotted with r)^2

This potential energy function also has a dependence on angular position, as it involves the dot product of the position vector r and the unit vector in the z direction. Therefore, angular momentum would not be conserved.

4. U = α/r + β(z(hat) dotted with r)^2

This potential energy function has both a dependence on angular position and angular velocity. Therefore, angular momentum would not be conserved.

In summary, only the first potential energy function would conserve angular momentum. The other three would not conserve angular momentum due to their dependence on angular position or velocity.