1D string on a cylinder and torus.

In summary, the conversation discusses the physics of a 1D string with fixed end points and how it applies to a string under tension and confined to the surface of a cylinder of radius r. The differences between the two scenarios are outlined, including the wavelength of the standing wave of lowest energy, the additional movements of the string, and the possibility of excluding motion to one dimension. It is also noted that the physics remains unchanged by translation along the cylinder's length and that the shape and velocity of the string can be used to predict future motion. The idea of a conserved quantity, possibly momentum along the length of the cylinder, is also mentioned. The conversation ends with a request for further thoughts on the topic.
  • #1

Spinnor

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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.
 
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  • #2
@Spinnor did you ever come to any more understanding on this?
 

1. What is a 1D string on a cylinder and torus?

A 1D string on a cylinder and torus refers to a theoretical concept in physics where a one-dimensional string is wrapped around either a cylinder or a torus (donut-shaped object) in higher dimensions. This concept is commonly used in string theory and is used to study the behavior of strings in different types of space.

2. How is a 1D string on a cylinder and torus different from a regular string?

A 1D string on a cylinder and torus is different from a regular string in that it exists in higher dimensions and is wrapped around a specific shape. Regular strings are usually studied in three dimensions and their behavior is not affected by the shape of the space they occupy.

3. What is the significance of studying a 1D string on a cylinder and torus?

Studying a 1D string on a cylinder and torus allows scientists to better understand the behavior of strings in different types of space and how they interact with each other. This can provide insights into the fundamental nature of the universe and contribute to the development of theories such as string theory.

4. How is a 1D string on a cylinder and torus relevant to real-life applications?

While the concept of a 1D string on a cylinder and torus may seem abstract, it has practical applications in fields such as cosmology and particle physics. Understanding the behavior of strings in different dimensions can help scientists make predictions about the behavior of particles and the structure of the universe.

5. Are there any experiments or observations that have been conducted to study a 1D string on a cylinder and torus?

Currently, there are no experiments or observations that directly study a 1D string on a cylinder and torus. However, scientists use mathematical models and simulations to study the behavior of strings in different dimensions and to make predictions about their properties. These theoretical studies are an important step in understanding the fundamental nature of the universe.

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