Motion of a String: Classical Point of View

[N.J.R]

first, I'm sorry that not be good at English.

I don't know motion of a string from a classical point of view.
Generally it deals with quantum mechanics.
But I want to know motion of a string from a classical point of view.
In the case of a very simple open string(spin with c velocity), one mode string, two modes string,...and a simple close string, one mode string, two modes string...etc.
I know that it's complicate. I just hope obtain some ideas or information(website, books, treatise... everything). Then I can calculate that with computer.

 

[turin]

If you're going to use a computer, then can you just consider the tension and mass density, or do you need more help? Maybe a first year physics text would help. "Physics: for Scientists and Engineers" 4th ed. by Serway has a very simplified basic consideration.

 

[N.J.R]

Thank you for helping me.
In fact, professor suggested studing for motion of a string a classical point of view.
Frankly speaking, I don't know what did he mean by that.
I thought 'what motion? And what I do calculate?'. Anyway he said that he already knows the solution and there are some treatise about that. But I found just one treatise ("mechanics of a close string in the tensor field"). It will be helpful. (I hope) But I still can't understand. The problem is that I don't know how to start in this subject just like a person who don't know how starting the computer.(if the person know that then no matter )
What I do calculate?

 

[turin]

Originally posted by N.J.R
What I do calculate?

I don't really have a good idea what your goal is, so I can only speculate/suggest. Calculate the displacement of every tiny portion of the string in time.

For example, a guitar string

It is "clamped" on both ends, which gives boundary conditions. If it is plucked, from the plucking action you can get the initial conditions (which are really just more boundary conditions, but in time rather than space). Then, the tension and mass density will determine the response of the string given these boundary conditions.

If you want to put coordinates on it:

You can say that the unplucked string lies along the x-axis. The portion of the string at the point x = x' is raised up to y(x') = y₀ (and this deforms the string into a somewhat "triangular" shape). Then, at time t = t₀, the string is released.

The "triangular" shape is the initial condition, that is, y(x',t₀) = y₀, and dy/dt = 0, and y(x,t₀) is a "triangular" function of x. Then, at the point x = x', the y position of the string will be y = y(x',t).

What will happen immediately after the string is released? Will y increase? No. Why not? Because the tension is pulling it in the direction of decreasing y and the initial dy/dt = 0. So how much tension? Well, that's a little complicated as it depends on the rigidity of the string (less rigid means more triangular, more rigid means more of a bowed shape initial condition). You can use trigonometry and first semester calculus to figure out what the tension would be at every point along the string. Qualitatively, the tension gives it the speed and the mass density limits the speed.

The wave equation looks like this:

(∂²y/∂x²) - (u^-2)(∂²y/∂t²) = (β)(∂y/∂t)

where u is the wave velocity along the string, and β is the damping coefficient due to air resitance.

For the string, this becomes:

(∂²y/∂x²) - (μ/T)(∂²y/∂t²) = (β)(∂y/∂t)

where μ is the mass density, and T is the tension. This is a linear, homogeneous approximation. In general, the tension should depend on the displacement, y, and the mass density should depend on the tension (by Young's modulus). Both are geneally interrelated as well in terms of a dependence on position.

 

[N.J.R]

I need your help again.
I read "D-Brane Primer"(by Clifford V. Johnson)
( http://arxiv.org/PS_cache/hep-th/pdf/0007/0007170.pdf )
That's what I want! But I confront with a mathematical difficulty. I don't know the expression. I have no idea what that means. So, will you please recommend me appropriate books to understand the notation and calculration? And in that treatise, at p.14(A Rotating Open String), I need a piece of advice(What's wrong with that?).

 

[turin]

Originally posted by N.J.R
That's what I want!

I don't understand.

Originally posted by N.J.R
I don't know the expression. I have no idea what that means.

What expression?

Originally posted by N.J.R
So, will you please recommend me appropriate books to understand the notation and calculration?

Well, I'm probably in the same boat as you are as far as knowing what books are good. I will read through the paper and see what notation is used. Then, after I've gotten a feel for it, I'll let you know if I can think of any books that would help you understand the notation. But, holy crap, this paper is 222 pages! That's one hell of a paper! Hopefully, someone who knows this stuff can come in here and set both of us straight.

Originally posted by N.J.R
... at p.14(A Rotating Open String), I need a piece of advice(What's wrong with that?).

What's wrong with what? I'll read page 14 right now to see if I can figure out what you're talking about.




EDIT

OK, I read the article up to page 14. I didn't notice anything wrong with it. I didn't read it with maximum scrutiny, but it seemed to all make some kind of sense. The author defines almost every variable and equation used in the paper up to that point.

Is the tensor notation confusing you?

For that, if you are not very comfortable with math, you might want to look into Ohanian's "Gravitation and Spacetime," although this is where I was introduced to tensor notation, and it confused the hell out of me (or maybe it was my major prof who confused me).

I am currently reading through a book written by Rindler called "Relativity: Special, General, and Cosmological." I haven't finished it yet, so I haven't yet had a chance to reread it to determine whether or not it would be a good source for gleaning specific contained concepts. Overall, though, I'm liking the way it is written.

Other than the tensor stuff, I didn't notice anything else that wasn't defined in the paper itself. Just in case you hadn't noticed, the very first two citations are for recommended reading for confused audience; at least, that's what I understood them to be. Check the citation list for those, and then see if you can get a hold of them.

If you post specific questions about specific equations or variables used in the paper, or even if you have questions about the prose, then just post them, and I, or hopefully someone more knowledgible, will attempt to give you a better understanding.

 

[N.J.R]

The book that you recommend is very nice. Nevertheless, I can't understand eq.(12),(27),(28) and configuration(that is X0,X1,X2) of p13,14.(I'm sorry that I don't know how to write the eq. in this post.) How so?

 

[turin]

Originally posted by N.J.R
The book that you recommend is very nice. Nevertheless, I can't understand eq.(12),(27),(28) and configuration(that is X0,X1,X2) of p13,14.(I'm sorry that I don't know how to write the eq. in this post.) How so?

First, I'm going to assume that you are referring to the Ohanian book. I have the first edition. Nevertheless, in which chapter are these equations? Are you talking about the paper that you have cited? Please try to be more thourough/specific with your questions in order to expedite the answers. I guess, at this point, I'll go ahead and assume you are referring to equations in the paper.

eq (12):
This is the action integral for the worldsheet, that is, instead of one parametrization over which to integrate to get the action, you now have two. So, instead of extremizing the length of a worldline, the area of the worldsheet is extremized. This gives the physical worldsheet that the string will sweep out in space time. The integral is a double integral (I'm pretty sure), so that you integrate over both τ and σ (thus, area as opposed to length). There are two different metrics involved here. The Minkowski metric is for proper displacement in spacetime. The new metric is a second rank tensor in τ-σ space, that is, it operates after τ has been found. The reason this new metric is needed, as I understand it, is that the parameter σ is not a proper length, and is in fact unitless by the way it has been defined.

eq (27):
There are many substitutions that you have to follow from equation (20). It is just varying the action with respect to the spacetime coordinates directly.

eq (28):
This is the result/consequence of extremization of the action (eq (27)) under space-time position variation (it gives the trajectory; it is the equation of motion). The middle equality is just the way to define the Laplacian.

 

[N.J.R]

Right you are. I told the paper what I have cited. But what I asked you are not that. I am poor at figures-specially, in this string theory. I know the variational principle, partial integration and Laplacian. Also, I read the Ohanian's book and the paper several times. Nevertheless I couldn't understand the process of calculation. Please will you give me a full explanation mathematically? (In fact, I don't know eq.(3), second line of eq.(13), eq.(19), (21), (22), middle line between (21) and (22), eq.(24), (25), (26), (29), (30), (41), (42), (45), (46) and (47) too. That's all that I wanted to know. It will be more trouble than a cartload of monkeys. So, you may explain the key principles to make myself understood.)

 

[lumidek]

Call For Votes

For those who asked how can they vote about the USENET newsgroup sci.physics.strings: we can only provide you with a link to the official description of the poll:

http://groups.google.com/groups?group=news.announce.newgroups&selm=1077593588.15146@isc.org

 

[selfAdjoint]

Lubos, I sent off my vote, YES, of course. I hope it helps. BTW instead of saying delete everything but the ballot, why not say copy paste just the ballot into a new mail within your email system, and then edit to make your vote and send it to the address they provide.