This is a followup to the previous discussion on starting a collaborative research project.

We've decided to start with a simpler (in the sense of being solvable) problems first, so here's a bunch of them and I'd like people to post an answer (and then up-vote it) for whichever one would they like to work on most (you can of course add a problem of your own). These problems are mostly taken from references Approximist has given last time.

  1. In a container filled with a liquid, heat transport will occur when the bottom of the container is heated and the top surface is cooled. How does the phenomenon change when the container rotates about its vertical axis?

  2. If you ride too slow, you fall over – as it is well known, if one rides the bicycle too slow, it is easy to capsize. Let us model this (in reality rather complicated) phenomenon by a single hoop of mass $m$ and radius $R$ (the mass of the hoop is concentrated on the very rim), which rolls vertically on a horizontal ground at velocity $v_0$ without slipping. What is the minimal value of $v_0$ , if this motion is stable against small perturbations? What is the frequency of the vibrations (waggling) of the hoop, if it is slightly pushed out of its original motion?

  3. Determine all the possible paths which a light beam can trace in the gravitational field of a spherically symmetric body. Does any closed path exist?

  4. Consider a one dimensional harmonic potential complemented with a power series beginning at a cubic term. How should one choose the all nonzero Taylor coefficients so that the exact period becomes independent of the amplitude? Try to guess the full potential function either from a few lower order coefficients, or, from other simple physical reasoning, and then show the constancy of the period.

  5. How far can an astronaut reach from Earth during his lifetime? Let us assume that such rocket technology is available which can provide constant acceleration for an arbitrarily long time! (The problems of rocket fuel and other disturbing details can be disregarded!) Let us choose reasonable parameters, such as 30 years of travel, acceleration equals to the terrestrial gravity! Also consider the recently mostly accepted Universe model with accelerating expansion (Lambda-CDM) with its presently known parameters!

  6. Determine the energy levels of an electron confined in a box with the sides of the box given as a and b, height h, and a homogeneous vertical gravitational force acting on the electron! The sides of the box can be regarded as infinitely high potential walls. Study the high energy (semi-classical) limiting case!

  7. A model for phase diffusion of a simple harmonic oscillator is provided by the master equation ${{\rm d} \rho \over {\rm d} t}= −\Gamma [a^{\dagger} a, [a^{\dagger} a, \rho]]$ Calculate the Fokker-Planck equation of the density matrix $\rho$ in $P$-representation. Determine the Langevin equations from the Fokker-Planck equation.

P.S.: As this is not really about the site itself, could it be posted on the main site?

P.P.S: also, it seems TeX isn't enabled here.

  • 1
    $\begingroup$ (1) IMHO, this can totally feature in the main site — i believe it would even increase the participation. (2) Marek, could you please edit your original post and put a link to the previous discussion? I think that, in this way, things stay 'self-contained', in an HTML-way. ;-) $\endgroup$ – Daniel Mar 7 '11 at 2:51
  • $\begingroup$ do we actually need MathJax enabled on meta? it's very VERY heavy. Like hundreds of kilobytes heavy. $\endgroup$ – Jeff Atwood Mar 7 '11 at 5:12
  • $\begingroup$ @Jeff: not at all. I mentioned that as another reason to move the question to main. $\endgroup$ – Marek Mar 7 '11 at 8:20
  • $\begingroup$ @Daniel: I believe I did link to that. Something must have gone wrong... Going to fix now, thx. $\endgroup$ – Marek Mar 7 '11 at 8:21
  • $\begingroup$ +1 for choosing some really "sexy" questions. I've been busy recently with exams, but hopefully I'll be ready when this kicks off. $\endgroup$ – Approximist Mar 13 '11 at 5:13
  • $\begingroup$ What happened with this? $\endgroup$ – user120404 May 22 at 17:04

(6) Also sounds 'sexy' and can be done using SciPy (or PyLab) to make the concepts very clear. Further, studying what happens to the solutions as 'b → a' (or vice-versa, depending on who's larger: no loss of generality here) may give insights into the symmetries of the problem and of its solutions — and may open a nice door to discussing about Hermitian Complements of Operators.

Just my 2¢…

  • $\begingroup$ Would you consider splitting this into two answers so that people can up-vote just one of them? By the way, I agree these are nice problems and I had a hard time choosing the favorite one myself. $\endgroup$ – Marek Mar 7 '11 at 8:31
  • $\begingroup$ @Marek: not a problem — done. $\endgroup$ – Daniel Mar 7 '11 at 15:46

[Following Marek's suggestion, i'm splitting my original answer in two.]

(3) Finding CTCs in Schwarzschild-like solutions (with or without angular momentum, ie, Kerr) is sort of a sexy problem — and, nowadays, can probably be done using Mathematica (or some other sort of symbolic computation software). In fact, this problem can possibly be readily generalized to that of the "Twin Paradox" in GR (as opposed to SR); which may lead to a nice discussion about homologies holonomies in GR (and their relation to curvature).


Personally, I'd like to work on (7). It's always nice to derive something useful from microscopic laws. Not that I know much about this stuff but I think this is a good opportunity to learn more.


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