Interesting solutions to classic physics problems

[Totally]

From time to time I hear about people coming up with creative/"non-mainstream" solutions to classical physics problems, whether by looking from a very different angle or using some unusual math that's unknown to anyone but that one slightly quirky professor from faculty of mathematics. However I have sadly yet to see one myself.

So, here is my question to professors, teachers, tutors and students of pf, do you have any such stories? Please share!

Edit: I was noted that my wording was ambiguous. By "non-mainstream" I mean a creative yet legitimate way that one is unlikely to find in most textbooks because it is less obvious or too much out-of-the-box. It was not my intention to discuss pseudoscience.

 

[Geofleur]

I really like getting particular solutions to ODEs by "dividing by" differential operators (as in, e.g., Hung Cheng's book on advanced analytic methods). It's equivalent to variation of parameters, but it's easier to work with in many cases. As an example of the method, suppose we want to find a particular solution to

$y'' + y = \sinh(x)$.

We rewrite this, using the notation $D = d/dx$, as

$(D^2+1)y = \sinh(x)$.

Now we divide by the differential operator (!) to get

$y = \frac{1}{D^2+1}\sinh(x)$.

Now note that the effect of $D^2$ on $\sinh (x)$ is to produce $\sinh (x)$ again. So $\sinh (x)$ is an eigenfunction of $D^2$ with eigenvalue 1. Hence, we replace $D^2$ with 1 everywhere we see it, to get

$y = \frac{1}{2}\sinh(x)$,

which can be seen to satisfy the original equation.

 

[Totally]

I love manipulations like this one. Reminds me of the first time I saw Lagrange multipliers - seemingly making stuff more complicated until you realize there is no need to actually calculate the λ, or in this case D². Going to try to use this one from now on

 

[mpresic]

While I was teaching Freshman physics, the students were assigned a problem concerning a fly who had to "fly" to the opposite corner of the room with given dimensions. It was a straightforward application of the Pythagorean theorem in three dimensions (or equivalently two applications of the Pythagorean theorem in two dimesions). Part b of the problem asked, if the fly loses its wings and must walk to the opposite corner, say from the corner where the floor, left lateral wall and front lateral wall meet and the corner where the right lateral wall, the back wall and the ceiling meet, what is the shortest distance he can walk.

One of my students was well versed in calculus and solved the problem correctly as a minimization problem. He described how far the fly would walk around the floor to the edge and up the back wall. I asked him why he did not "unfold the room". I showed him a shoebox for that purpose. The shoebox could be unfolded and then it became obvious that the answer was just a straightforward application of the Pythagorean theorem (with appropriate lengths).

I was a grad student at the time and Resnick,(the author of the physics textbook the problem came out of) was also teaching a similar class. When I told him I taught my students to unfold the room, he acted like, are you all still surprised by that old chestnut. Unfolding the room is obvious. It turns out I got the answer from my childhood. My grandfather had a set of 1926 Book of Knowledge encyclopedias with this puzzle (ostensibly for children). Resnick was about 30 years older than I and no doubt he probably saw the answer in his childhood as well (Even Resnick was a small child in 1926).

 

[Ben Niehoff]

There's a very old method of constructing architectural structures that was employed a lot by Gaudí. Take a string (or even a network of strings) with various loads attached, and suspend it from some anchor points. An ideal string carries only tension. Let the entire structure hang in equilibrium. Now imagine replacing every segment of string with a rigid material so that the entire structure holds its shape. Turn the entire thing upside-down, and now every segment experiences pure compression with no lateral forces. You now have a tower or arch that can support the desired loads and can be built out of concrete.

http://dataphys.org/list/gaudis-hanging-chain-models/