Frenet–Serret formulas, what are the possible applications?

[andonrangelov]

Hi all,
I know that Frenet–Serret formulas are:
dT/ds=kN
dN/ds=-kT+fB
dB/ds=-fN
and also that this formula effectively defines the curvature and torsion of a space curve (http://en.wikipedia.org/wiki/Frenet–Serret_formulas), but:
1) Where else one can find those equations?
2) Do you have an idea for possible applications?
3) What physical systems obey those equations?

 

[Studiot]

Not long ago this question was asked

https://www.physicsforums.com/showthread.php?t=483983

The thread was never developed since the OP never came back but if you apply those formulae to such roofs, power station cooling towers and the like you will notice certain special characteristics. Hull shapes in Naval Architecture is another place where they have application.

The surfaces of such shapes are developable surfaces so they can be easily and accurately defined, drawn and fabricated.

Every continuous space curve obeys these formulae, only some lead to developable surfaces.

 

[andonrangelov]

Thanks for this interesting example that you pointed in Architecture, but does someone knew more physical examples of those equations?

 

[Bill_K]

They are often used in connection with the Thomas Precession in special relativity. That is, the precession of a gyroscope being carried along a curved path.

 

[andonrangelov]

Does someone know other applications, or where this formulas may appear?

 

[Abel Cavaşi]

I found a theoretical application of these formulas and I called those results "The recurrence theorem of the Frenet formulas".

I find it strange that no talks more about these extremely important formulas that meaningful objective, independent of the Cartesian benchmark that defined the trajectory of a body.

Must be something more to this formulas and I'm glad you still looking for their essence.

 

[andonrangelov]

Abel, thanks for your comments. I am interested in those equations because they appear in different physical phenomena like three state quantum system, Lorentz force and many more… Normally there is a contraintuitive solution for those equations and I want to see what will be the result in different area of physics.
Here we have find interesting solutions of this equation http://arxiv.org/abs/0812.0361
Thus if you know some physical cases where this equation appear then please give directions.

 

[Abel Cavaşi]

Unfortunately, because I'm not familiar with the strange quantum mechanics, I did not understand too much of your article to which you referred.

But, be pointed out that in the recurrence theorem of the Frenet formulas occurs very frequently the ratio between curvature and torsion (parameter that we can call "lancretian", from Lancret's Theorem). It also occurs very frequently the root of the sum of the squares of curvature and torsion, which is exactly the Darboux vector module (which is why we call this important parameter as "darbuzian").

Now notice that in the study of complex numbers occurs very frequently the ratio between the real part and the imaginary part, and also the complex number module. For this reason, I think would be interesting to admit that to an any curve we can associate a complex number q that have the torsion on the real part and the curvature on the imaginary part, ie
$$q=\tau+\textbf{i}\kappa$$,
number that we can call it "complex torsion".

Thereby, we introduce the complex numbers in the theory of curves and we can take the advantages of results obtained in complex analysis.

Therewith, I also recommend the study of closed curves and especially of closed helices, because of the possibility that an elementary particle be just a luxon moving on a closed helix.