Conservation law associated with the symmetry of a helix

[Hiero]

In a problem in Landau’s mechanics (end of section 9) he asks for the quantity conserved in the field of “an infinite homogenous cylindrical helix.”

The solution is that the Lagrangian is unchanged by a rotation of dΦ together with a translation of hdφ/(2π) (about and along the symmetry axis) where h is the pitch of the helix. This implies that M_z+hP_z/(2π) is conserved where M and P are the angular and linear momenta and _z means the component along the symmetry axis.

It’s a very nice example of Noether’s theorem, but I have one question:

Is he silently assuming that the helix is right handed? Surely for a left handed helix the conservation law would be M_z-hP_z/(2π), right?

Just want to make sure I’m understanding that correctly because Landau never mentions the orientation of the helix, but I think it matters.

Thanks.

 

[Orodruin]

A left-handed helix would have a negative $h$.

Edit: In other words, what matters is the definition of the symmetry as the transformation of $d\phi$ in the angle being accompanied by the translation $h\, d\phi/(2\pi)$.

 

[Hiero]

A left-handed helix would have a negative $h$.

When I looked up “helix pitch” I just found that it was the step height, but I guess some people also encode the orientation like this.

Thanks Orodruin!