What is Conserved in a Homogeneous Infinite Cylindrical Helix?

[Wumbate]

I'm having trouble with this problem:
Find what is conserved in space where an infinite cylindrical helix is homogeneous.

It comes from Mechanics by Landau and Lif****z, page 21, solution on page 22. Here is a google books link to it
http://books.google.com/books?id=Lm...ig=iiqa9SIPmqtlKY3nimsuLjBqCbM&hl=en#PPA22,M1
(the solution is at the top of the linked page 22, part h)

I don't see why partial L/partial phi is dM_z/dt..

 

[Jhero]

Hello,

Thank you for bringing this problem to our attention. In order to better understand the solution provided in the book, let's break down the problem and discuss the concept of conservation in space.

First, let's define what we mean by "conservation in space." In physics, conservation refers to the principle that certain physical quantities remain unchanged or constant over time. This means that these quantities are conserved, or preserved, in a given system. In classical mechanics, there are several quantities that are conserved, such as energy, momentum, and angular momentum.

Now, let's look at the specific problem you have mentioned. The problem describes an infinite cylindrical helix that is homogeneous, meaning that its properties are the same at all points along its length. The problem asks us to find what is conserved in this system.

To solve this problem, we can use the principle of conservation of angular momentum. In this case, the angular momentum of the system is conserved because there are no external forces acting on it. This means that the total angular momentum of the system remains constant over time.

In the linked solution, the book uses the Lagrangian approach to solve for the conserved quantity. The Lagrangian, denoted as L, is a function that describes the dynamics of a system. It is defined as the difference between the kinetic and potential energies of the system. In this case, the Lagrangian for the cylindrical helix is given by L = T - U, where T is the kinetic energy and U is the potential energy.

The book then takes the partial derivative of the Lagrangian with respect to the angle phi, which represents the rotation of the helix. This partial derivative, denoted as dL/dphi, represents the angular momentum of the system. This is the quantity that is conserved in this system.

Now, to answer your question, the book uses the fact that the angular momentum is conserved to solve for the time derivative of the angular momentum, dM_z/dt. This is done by taking the time derivative of the Lagrangian and setting it equal to the torque acting on the system, which is given by M_z. By doing this, the book shows that the time derivative of the angular momentum is equal to the torque, which is dM_z/dt = M_z. This is why the book equates the partial derivative of the Lagrangian with respect to phi, dL/dphi,