1D string on a cylinder and torus.

[Spinnor]

The physics of a 1D string with fixed end points is found here:

http://www.uio.no/studier/emner/matnat/ifi/INF2340/v05/foiler/sim04.pdf

Now imagine a string under tension T and of mass density rho confined to the surface of a cylinder of radius r. I posit that this string will act just as a string with fixed end points does with the following differences:

1. The standing wave of lowest energy will have wavelength 2*pi*r whereas the standing wave of lowest energy for a string of fixed endpoints will have wavelength L/2.

2. In addition to vibrating the string can also move along the cylinder and the string can rotate around the cylinder.

3. As with a string with fixed end points we can exclude motion of a small piece of the string to only one dimension (though we don't have to)

As the physics is unchanged by translation of the string along the cylinder's length we will have a conserved quantity? Momentum along the length of the cylinder?


Let us identify the ends of the cylinder. We can now label displacements of the string with an angular coordinate theta. Picture this string in some simple combination of vibration and translation. Stop time. Knowing the shape of the string along with the instantaneous velocity for each point of the string is all one needs to predict future motion. The physics of the string does not change if a constant is added to theta.

Thanks for any thoughts.

 

[Greg Bernhardt]

@Spinnor did you ever come to any more understanding on this?