16 year old solves 300 year old problem set by Isaac Newton

D H

Corrected.

My main point still stands: Wolfram alpha (Mathematica) occasionally makes some absolute howlers. It assumed [itex]\frac{du}{da}[/itex] meant [itex]\frac{\partial u(k)}{\partial a}[/itex]. It then assumed that since u is a function of k that this means [itex]\frac{\partial u(k)}{\partial a}=0[/itex]. That is a howler.

K^2

That's purely Alpha. It does best it can to interpret the input. How the heck is it supposed to know what u is a function of? In Mathematica, you would have to enter it explicitly.

This way, there can be no ambiguity. But there is no room for user error, either. If you put = instead of ==, Mathematica isn't going to try and guess what you meant. It will actually treat that whole expression as 0 from there on, because that's what your code requested. Alpha tries to be peasant-friendly, so it will obviously resolve ambiguities in favor of simplicity.

Kholdstare

Did he do something along this line?

http://www.sciencedirect.com/science...20746201000683

HallsofIvy

What really seems strange to me is that the solution to a problem proposed by Newton and unsolved by mathematicians and physicists for 300 years would win second prize in a local school competition. I want to see the paper that won the first prize!

gturner6ppc

Someone thankfully found more on the solution Ray found, including the ongoing discussion of the derivation, including Maple code, but I can't directly link it here till I post 10 comments, so Google

teen_solves_newtons_300yearold_riddle_an/c4sxd91

and the discussion thread at reddit will be the top hit.

The solution is kind of simple once you see it. ^_^

lugita15

It seems that the first prize was given to a student who supposedly solved the problem of relativistic ray-tracing. It seems that either there are a lot of groundbreaking developments going on that no one's heard about, or the German competition judges are not judging very well, or the media is blowing things way out of proportion.

Dickfore

I'm for this choice.

Ayre

On reddit:

http://www.reddit.com/r/math/comment...r_calculating/

(Sorry, can't post links, add "http")

there's a link to a picture of him holding up a particular formula. This seems to be a constant of the motion for a projectile moving in uniform gravity and quadratic drag.

As is pointed in the comments thread, that particular formula is (a) easy to derive, (b) known since at least 1860.

On reddit, this particular formula seems to be taken to be the full extent of his solution. However, I don't think that's true.

There's a picture of him standing in front of his poster:

http://www.jugend-forscht-sachsen.de...dex/image3.jpg

The section he's pointing at seems to have the title "Lösung", so the two equations there are presumably his solution.

You can see that
(i) the solution involves two formulae
(ii) they're both fairly long.
(iii) they're both of the form LHS = numerator/denominator.

The particular first integral linked on reddit appears just to the right of his hand. Apart from the actual solution equations, this is the only boxed equation visible on the poster. Thus, he clearly considers that equation important.

Also of interest is the top part of the poster, which seems to be a historical review of the problem. With some intelligent guessing, one can problably work out some of the capitalized names, there. It might be interesting to know to what extent Mr Ray was aware of previous work.

Pure speculation below: This boy has reduced the problem to quadrature, and as an important first step, found a particular first integral to the equation. Finding such a first integral does require ingenuity, for which the boy should rightly be proud, but alas, it's not new.

Depending on "analytical solution", reducing a differential equation to quadrature might or might not count. This ambiguity led someone, somewhere, believe that Mr Ray found something genuinely novel, and then it only take the combined hysteria of the world's news outlets to blow it way out of proportion.

As HallsofIvy, the fact that he only won second prize doesn't really seem compatible with finding a solution, to an important problem, which has eluded physicists and mathematicians for 300 years. Perhaps the jury was well aware that Mr Ray's feat was impressive, but not quite as groundbreaking as the media seem to portray.

Ayre

I found a high-res picture of the poster image!

http://i47.tinypic.com/2v0oco8.jpg

So his solution is

[tex]
u(t) = \frac{u_0}{1 + \alpha V_0 t - \tfrac 1{2!}\alpha gt^2 \sin \theta + \tfrac 1{3!}\left(\alpha g^2 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots}[/tex]
[tex]
v(t) = \frac{v_0 - g\left[t + \tfrac 1{2!}\alpha V_0 t^2 - \tfrac{1}{3!}\alpha g t^3 \sin \theta + \tfrac 1{4!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^4 + \cdots \right]}
{1 + \alpha V_0 t - \tfrac 1{2!} \alpha gt^2 \sin \theta + \tfrac 1{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^3 + \cdots}[/tex]

Thus, he has found the velocity in terms of Taylor series in time. Nothing revolutionary, in other words.

The poster also claims that the constant of the motion he found is a "fundamentale neue Eigenschaft", but as pointed out in the reddit thread, it has been known since at least 1860.

Of course, I don't blame him. What he did was very impressive, but hopefully he has learnt to be more careful before claiming new solutions to old problem.

Hurkyl

How similar were the claims he made to what the media made?

Ayre

Well, they're certainly not as hyperbolic.

Certainly, his poster claims that he has discovered something new. It says, for instance, ""erstmals vollanalytische Lösung eines lange ungelösted Problems", i.e. "first fully analytical solution of a long unsolved problem" (my translation). I guess there was a misunderstanding, and he didn't realise that when people say that no analytical solution of this problem has been found, series solutions do not count.

Of course, for all we know, the boy himself knows the merits of his work very well. It might well just be a parent who pushed him to use more grandiose language in his poster than was justified.

Also this poster only shows one of problems he solved. It could, of course, be that the work on the other problem is truly groundbreaking. But I have my doubts.

Kholdstare

Thank You very much Ayre. I am glad that you shred some lights on what he actually did. Without, that verification, we can not give him any credit.

I see media has misunderstood, as sometimes seen for scientific journalism. (Or maybe he made wrong claim - although chances are low). Probably the best title would be "First analytical series solution ..."

Euler1707

What are the assumptions for the drag of this problem? -kv^3 ?? squared?? Also, I have read that this has something to do with an object bouncing of a wall. Hmmm can anyone post the original problem in differential equation form?


Maybe I can solve it today and create another Newton-Leibniz controversy. I am Mexican so it would be awesome.

Ayre

The problem is that this is most certainly not the first analytical series solution. If I have time later, I might go hunt for some references, but solving differential equations by series has been part of the standard toolbox for a very, very long time.

It could be, to give him the benefit of the doubt again, that his particular series solution have practical advantages over other series solutions. For instance, his series might be converging really rapidly, providing thus more accurate algorithms than was possible before.

But this is not something he's claiming, as far as I can tell. And again, it's very unlikely that he's stumbled onto something nobody has thought of before. Back in the 18th and 19th centuries, a lot of people were very busy hurling artillery rounds at each other; presumably, without computer, calculating artillery tables was a laborious task, and much work would have gone into finding good practical methods of solving the differential equations.

There are two distinct problems.

The first problem is to find the motion of a point particle traveling in uniform gravity, with drag proportional to the square of its speed. The governing equations, as they appear in the poster I linked earlier, are

[tex]\dot u(t) + \alpha u(t) \sqrt{u(t)^2 + v(t)^2} = 0 \\
\dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g[/tex]

Here, u and v har the horisontal and vertical components of the velocity of the particle, g is the gravitational acceleration, and alpha is a coefficient of drag, so that the deceleration due to drag is alpha*(u²+v²). The initial conditions are that v(0) = v_0 > 0, and u(0) = u_0 ≠ 0.


The second problem seems to be about particle collision. I haven't found any information on this beyond the German abstract posted earlier in the thread, nor do I know enough about the subject to make an intelligent guess.

m2840

In 2010 I have published (Military Academy of Lisbon) a little paper on the problem, with the drag force quadratic in the speed, and solve it numerically for a standard rifle projectile.
My paper:
Ferreira, R. (2010). Movimento de um Projéctil de G3: Força de Arrasto no Quadrado da Velocidade. in Proelium – Revista da Academia Militar, VI Série, nº 13.

When the paper was almost done, I discovered a 2007 paper that claims to have found an analytic solution. Here it is the reference:
Yabushita, K., Yamashita, M. & Tsuboi, K. (2007). “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method”. Journal of Physics A: Mathematical and Theoretical, Vol. 40, pp. 8403–8416.

Ayre

Good catch!

This reference also appears on Mr Ray's poster, so he must have been aware of it. Though ge describes it as a "semianalytische exakte Lösung", or a "semi-analytical exact solution".

I only glanced through the paper (link), but it seems to only give power series solutions, together with recursive formulae for the coefficients. So these authors also seem to be using the term "analytical solution" as something distinct from "closed-form solution".

Furthermore, Mr Ray's solution appears to have terms of type similar-looking to the ones in this paper. Perhaps he did something similar, modified it in someway promoting it from "semi-analytical" to "analytical"?

Euler1707

Very nice, I have seen similar problems. I posted here once that I could not solve a trivial problem where the drag varied as the cube of the velocity.

The only way I could solve it was using a Taylor expansion. I plotted but the solution curved seemed a little odd. I could not find a physical justification for such peculiar curve.

I would love to read your paper.