350-Year-Old Newton's Puzzle Solved By 16-Year-Old

First time accepted submitter johnsnails writes "A German 16-year-old, Shouryya Ray, solved two fundamental particle dynamic theories posed by Sir Isaac Newton, which until recently required the use of powerful computers. He worked out how to calculate exactly the path of a projectile under gravity and subject to air resistance. Shouryya solved the problem while working on a school project. From the article: 'Mr Ray won a research award for his efforts and has been labeled a genius by the German media, but he put it down to "curiosity and schoolboy naivety." "When it was explained to us that the problems had no solutions, I thought to myself, 'well, there's no harm in trying,'" he said.'"

Comments

Re:I thought these were pretty much known already

[Anonymous Coward]

There is no problem solving the equations numerically. This kid found analytical solutions to the equation of motion (or at least, that's how I read TFA). Punching in the exact solution is faster and more accurate than taking a zillion small but discrete steps, which is what you're stuck doing right now. Well, that depends on the complexity of the solution, but as a general rule...

When in Doubt...

[Rie Beam]

...go to the source! The German articles I've scoured seem to have a little more information about the problem itself and what he actually accomplished. The oldest one only records that he "claims" to have solved them (earlier this month), but so far no actual data. Close.

http://www.enso-blog.de/jugend-forscht-drei-arbeiten-aus-ostsachsen-beim-bundeswettbewerb
http://www.morgenpost.de/vermischtes/article106358144/16-jaehriger-Schueler-loest-uraltes-Mathe-Problem.html

Gotcha!

[Rie Beam]

http://jugend-forscht-sachsen.de/2012/teilnehmer/fachgebiet/id/5

Text is in German. It all stems from a Youth Research competition he entered into back in March of this year. This is, so far, the best summary I've found -- there is a paper, apparently, but no link just yet.

'Two problems in classical mechanics have withstood several centuries of mathematical endeavor. The first problem is therefore to calculate the trajectory of a body thrown at an angle in the Earth's gravitational field and Newtonian flow resistance. The underlying power law was discovered by Newton (17th century). The second problem is the objective description of a particle-wall collision under Hertzian collision force and linear damping. The collision energy was derived in 1858 by Hertz, a linear damping force has Stokes (1850) is known. This paper has so far only the analytical solution of this approximate or numerical targets for the problems solved. First, the two problems are solved fully analytically. For the first problem will be investigated further using the analytical solution, the physical behavior of the system and set up outline solutions for generalized models. For the second problem is carried out in order to increase efficiency and convergence control a semi-analytical optimization. Finally, the analytical results are compared with numerical solutions so as to validate accuracy and convergence to numerically."

Re:Gotcha!

[bcrowell]

That helps a little, but still doesn't really clarify completely what he did. I'll explain a little about what I know about the projectile problem and what I can figure out about what he might have accomplished here.

In the Principia, Newton poses three closely related problems. One is projectile motion under the influence of a frictional force that's proportional to velocity (book II, section I). Next he considers the case where the friction is proportional to the square of the velocity (book II, section II), and finally the case where it's of the form av+bv^2, where a and b are constants (book II, section III). Let's call these cases 1, 2, and 3.

Case 1 is pretty straightforward. The x and y motions are decoupled, and each of the motions is governed by a first-order, linear, inhomogeneous equation.

Case 2 is actually of more physical interest than case 1 for most real-world projectiles. For example, when you toss a baseball in air, its Reynolds number is about 10^4 or 10^5, and in that regime, a force proportional to v^2 is a pretty decent approximation. There is a well known closed-form solution for the one-dimensional subcase (I actually had a student a few years back who figured it out for herself, which was impressive), which is y=A ln[cosh(t sqrt(g/A))].

A hint is that this page has a photo of him holding up a large sheet of paper with his closed-form solution on it. The equation is clearly visible, and reads g^2/(2u^2)+(alpha g/2)[v sqrt(u^2+v^2) / u^2 + arsinh |v/u|] = const. The notation isn't explained, but clearly u and v are the components of some vector, probably the velocity vector. If so, then the constant alpha has to have units of inverse meters.

This makes me think that what he's solved is the full two-dimensional version of case 2. It can't be case 3, because besides g there is only the one constant alpha appearing in his equation. If you write down the equation of motion, a=F/m=(mg-bv^2)/m=g-(b/m)v^2, the constant that naturally occurs is b/m, which has units of inverse meters. It also makes sense that his solution has a hyperbolic trig function in it, since the y(t) for the one-dimensional version of case 2 has a hyperbolic trig function in it.

If my interpretation is right, then you should get a correct one-dimensional result from his equation when u=0. Unfortunately his equation blows up to infinity in that case, so I'm not sure how to extract any sane interpretation from it. By setting alpha=0, you should also get the case with zero friction. That does sort of make sense, since it says u is a constant, which it should be in that case.

It would be interesting to see if my interpretation is right by doing a numerical simulation and seeing if his expression really does seem to be a constant of the motion.

One thing to point out is that he may not have actually solved the full problem as set by Newton. He hasn't found the equation of the trajectory in closed form (which I think was what Newton was most interested in), and he also hasn't found the position in closed form as a function of time. (This is all assuming my interpretation is right.)

Re:Gotcha!

[Anonymous Coward]

You're right, he's demonstrated a constant of motion (i.e. a first integral) in the 2D version of Newton's Case 2. The constant alpha in his equation is what you called b. Gravity points in the -v direction.

You can easily check this by differentiating his equation with respect to time, and then eliminating the derivatives of u and v using the expressions

du/dt = -b u sqrt(u^2 + v^2)
dv/dt = -b v sqrt(u^2 + v^2) - g

His solution can probably be extended to Case 3 quite easily, if anyone feels like a challenge :)

Re:That Moment

[chill]

Germany still produces some rays of light.

To be accurate... he was born in India and moved to Germany with his family at age 12. He did not speak a word of German when he arrived.

While credit must be given to the German school system, I think most of his accomplishment comes from him and possibly his family.

Re:Specifics?

[HeLLFiRe1151]

This is an article from 1983. I believe it explains the problem.

http://www.annualreviews.org/doi/pdf/10.1146/annurev.fl.15.010183.000245

Re:I thought these were pretty much known already

[Anonymous Coward]

You forgot a lot of things:
-gravity is not a constant vector force downward. It is a radial force inward toward the center of the Earth, and its intensity varies with altitude.
-air resistance is not constant either. It depends on air pressure which varies with altitude as well.
-air resistance is not perfectly proportional to v^2, especially at transonic and supersonic speeds.
-if the projectile is spinning, it may cause a net aerodymamic force in a direction other than -v. Like a curveball.
-the earth is a spinning frame of reference, which results in various annoying effects.
-the air is not necessarily stationary. Wind exists.
and so on.

But we don't know whether this dude accounted for any of this stuff or not, because the goddamn article doesn't tell us.

Re:terrible article

[ObsessiveMathsFreak]

You are right. This article is awful, conveying no sense of the nature of the problem or its complexity, and giving no idea of the solution at all.

The only equations I'm aware of for a falling particle subject to air resistance take the form

m v' = -mg -a*v-b*v^2

which is a constant coefficient Riccati differential equation for the velocity v. I'm reasonably sure this would have an analytic solution.

Maybe complications arise in the 2D motion case, or perhaps the problem includes a particle which is also spinning. Maybe the drag terms take more complicated forms. I don't know. The article is pretty dreadful to be honest.

Re:terrible article

[Smurf]

That's "Analytische lösung von zwei ungelösten fundamentalen Partikeldynamikproblemen" or, in English, "Analytical solution of two fundamental unsolved problems of particle dynamics".

But that doesn't seem to be a paper published in a peer-review journal, but rather the title slide of a presentation he gave on March 1, presumably when when he received the Jugend Forscht ("Young Researchers") award.

And the kid is Indian, not German (as long as we can tell from the article).

And this is a problem in Physics, not in Mathematics. It shocks me that people get that mixed up.

And the kid looks 30 years old, but I would never hold that against him.

Re:Fermat & Poincaré

[2.7182]

Well, that is true but then after those papers appeared there was a several year effort by 3 groups to fill in the details and make it more digestible. Each of the resulting books/documents are several hundred pages long.

Some problems just require longer proofs.

Re:terrible article

[Anonymous Coward]

The one dimensional equation given does have an analytic solution (and in fact it isn't very hard, just a little intricate to integrate).

As you rightly suggest, it is the two dimensional problem that is a lot harder. As far as I know there is no exact solution; though perhaps Mr. Ray has found one. Indeed, Herman Goldstine in his magisterial "The Computer from Pascal to von Neumann" states that the reason why Americans during the war worked on computers was primarily to find solutions to this problem, so that artillery could be properly aimed.

Re:Explain the mind of a genius?

[the gnat]

...it does publish great papers, but does require something of a personal connection to get into... Same for The Proceedings of the National Academy of Sciences

Actually, this isn't so true of PNAS any more. One of the previous editors decided in the late 1990s to raise the quality prestige of the journal by accepting more papers through a traditional peer-review route, as opposed to NAS members "communicating" or "contributing" articles (which would often have minimal peer review). This was very successful, and now most articles in PNAS get in through the front door, and they're slowly eliminating the back doors. The overall quality is pretty good - not as high-impact as Science or Nature or some of the top specialty journals, but it's definitely a journal that researchers are excited about publishing in if they can't get into the top tier. The fact that they're not part of Elsevier or one of the other big commercial publishers, and their open-access fee is very reasonable, is an added bonus. (Disclaimer: I've published there, so I'm not entirely unbiased.)

Now, as with any journal, knowing the right people always helps - sadly, this is true at any level.

Re:That Moment

[tixxit]

The article states the father taught him calculus when he was 6. However, his father also says the kid passed his understanding a while ago and he doesn't understand the math used to solve this problem. Seems like the father was responsible for instilling a curiosity and some foundations, but after that it's all just this kid. You gotta give him credit.

Re:terrible article

[jordan314]

I was pretty disappointed that Slashdot wouldn't find the equation for this. I ended up finding it on reddit: http://www.reddit.com/r/worldnews/comments/u7551/teen_solves_newtons_300yearold_riddle_an/c4sxd91

Re:Flash journalism

[FrangoAssado]

Since we're linking to comments from Reddit: people also found out that this solution was known since at least 1860, and was published in a modern journal in as recently as 1977.

It's great that a 16 year old discovered this, and it could have been a cute (but not as flashy) story. But the reporter didn't even bother to talk to someone familiar with the field.

Re:That Moment

[galaad2]

I'd reserve your hosannas until this kid's magic formula gets published, along with a formal statement of the problem.

the formula has already been published, here: https://www.jugend-forscht.de/images/1MAT_67_download.jpg
(photo of the formula taken on May 18th)

article source:
https://www.jugend-forscht.de/index.php/projectsearch/detail/6038.4568
and
http://www.jufo-dresden.de/projekt/teilnehmer/matheinfo/m1

i can't find the full paper yet though, but on reddit some users claim that the formula works in Maple
e.g.
http://www.reddit.com/r/worldnews/comments/u7551/teen_solves_newtons_300yearold_riddle_an/c4szejb

where f is constant on the path the particle makes in the space of velocities:
f:=(g^2 /(2*u^2 ) + a*(g/2)*(v*sqrt(u^2 +v^2 )/(u^2 ) + arcsinh(v/u)));