Conservation Laws for Particles in Circular Cylinder

[Magister]

Homework Statement

A particle moves in a conservative field of force produced by a mass distribution. In each instance the force generated by a volume element of the distribution is derived from a potential that is proportional to the mass of the volume element and is a function only of the scalar distance from the volume element.
Supose that the mass is uniformly distributed in a circular cylinder of finite length, with axis along the z axis. What are the conserved quantities in the motion of the particle?

Homework Equations

The fact that if the space is uniform then the linear momentum is conserved and that if the space is invariant to rotations then the angular momentum is conserved.

The Attempt at a Solution

The linear momentum is not conserved in any direction and the angular momentum is only conserved along the z axis. Am I wrong?

But let's now supose that the cylinder has infinite length. In this case the space is uniform along the z axis and hence the linear momentum along z is also conserved. Am I wrong?

Thanks for your time.

 

[Dick]

I think your reasoning is correct.

 

[Magister]

What about if the mass is in the form of a uniform wire wound in the geometry of an infinite helical solenoid, with axis along the z axis?
The only symmetry in the system is a translation together with a rotation along the z axis. What does that means in terms of conserved quantities?