Does there exist a transformation between a line and a closed loop ?

[peter308]

Dear All:
For a line structure(say a very long atomic chain) and a closed loop structure( connect the head and tail of this atomic chain). Does there exists a transformation between these two structures?
For example if we want to study the vibration mode of these two cases. If we already know all the modes of one case then are we able simultaneously know the information of the other modes without any further calculation?

Best Yen

 

[Simon Bridge]

That's quite broad... a line of atoms, say 100 carbon atoms (centane?) would be a floppy chain able to wiggle about a lot. A loop of same would also be floppy. But hexane would be different from cyclohexane since the ring is under tension.

The vibrational modes at atomic levels work somewhat differently from classical harmonics.

On the large scale you may want to compare normal modes on a length of pipe with those on the same length pipe but in a circle - then it is just the boundary conditions that are different. (Though frequency will depend on wave speed as well - which depends on the tension in the pipe, which depends on how the pipe got to be a hoop - the wavelengths are not affected.) With everything else being equal - the change in boundary conditions is what determines the change in modes - and that is predictable.

You should be able to work that out for yourself.

 

[peter308]

Thanks For Mr.Simon's reply. In my case the line is a perfect straight line. I can derive a general form for the Irreducible representations of the vibration modes for any number of atomic chains ( with atom numbers N). I want to find out is there a way to transform the irreducible representations(vibration modes) for a straight line into the irreducible representations of a circle? So i don't need to calculate the irreducible representations of a circle. In fact the calculation of irreducible representations of a infinite large circle is a little troublesome. That is why i would like to know if any of this kind of transformation exists? With Best Regards
Yen

 

[Simon Bridge]

In my case the line is a perfect straight line.

So you are dealing with an idealization rather than actual chains of atoms?

In fact the calculation of irreducible representations of a infinite large circle is a little troublesome.

In what way is the circle infinite?
You were talking about a circle if N elements before.

That is why i would like to know if any of this kind of transformation exists?

I believe I've told you how to figure that out. Unless I've misunderstood the question.
It's a question of looking at the boundary conditions - some of the modes on the line are not going to be present on the circle. It is straight forward to work out which ones.

 

[HallsofIvy]

What do you mean by "transformation"? There exist a "function" between any two sets. There exist a bijective function between any two sets that have the same cardinality (as do a line segment and a circle). Since the line segment has two "boundary points" and a circle has none, there does NOT exist a continuous mapping one onto the other.