# Conservation law associated with the symmetry of a helix

• Hiero
In summary, Landau's mechanics problem regarding the conservation of a quantity in the field of an infinite homogeneous cylindrical helix can be solved by using Noether's theorem. 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 Mz+hPz/(2π) is conserved where M and P are the angular and linear momenta. However, the conservation law would be Mz-hPz/(2π) for a left-handed helix. Therefore, the orientation of the helix does matter in this case.
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 Mz+hPz/(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 Mz-hPz/(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.

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
Orodruin said:
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!

## What is a conservation law associated with the symmetry of a helix?

A conservation law associated with the symmetry of a helix is a principle that states that the shape and structure of a helix remains constant over time, regardless of external forces acting upon it. This is due to the inherent symmetry of a helix, which allows it to maintain its shape and structure.

## How does the symmetry of a helix contribute to conservation laws?

The symmetry of a helix plays a crucial role in conservation laws by providing a stable and predictable structure. This allows for the conservation of energy and momentum, as well as the preservation of other physical quantities, such as angular momentum and charge.

## What are some real-life applications of conservation laws associated with the symmetry of a helix?

Conservation laws associated with the symmetry of a helix have many practical applications, such as in the design of structures that need to withstand external forces, such as bridges and buildings. They are also important in fields such as biology, where the helical structure of DNA is crucial for the conservation of genetic information.

## How are conservation laws associated with the symmetry of a helix related to other physical laws?

Conservation laws associated with the symmetry of a helix are closely related to other fundamental physical laws, such as Newton's laws of motion and the laws of thermodynamics. They provide a framework for understanding how energy and other physical quantities are conserved in various systems.

## Are there any exceptions to conservation laws associated with the symmetry of a helix?

While conservation laws associated with the symmetry of a helix are generally considered to be universal principles, there are some exceptions. For example, in certain extreme conditions, such as at the quantum level or in the presence of strong gravitational forces, these laws may not hold true. However, they still provide a valuable framework for understanding and predicting the behavior of physical systems.

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