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

[surajt88]

mobile.news.com.au/breaking-news/world/german-teen-shouryya-ray-solves-300-year-old-mathematical-riddle-posed-by-sir-isaac-Newton/story-e6frfkui-1226368490521

Shouryya Ray worked out how to calculate exactly the path of a projectile under gravity and subject to air resistance, The (London) Sunday Times reported. The Indian-born teen said he solved the problem that had stumped mathematicians for centuries while working on a school project.

m.heraldsun.com.au/news/breaking-news/german-teen-shouryya-ray-solves-300-year-old-mathematical-riddle-posed-by-sir-isaac-Newton/story-e6frf7k6-1226368490521

Mr Ray has also solved a second problem, dealing with the collision of a body with a wall, that was posed in the 19th century.

I am still trying to figure out what the original problem was. Any thoughts on this?

 

[leroyjenkens]

I find it hard to believe a problem like that "stumped" mathematicians (and physicists too, I guess) for this long, only to be solved by a 16 year old kid.

 

[AlephZero]

Whatever it was, it won him 2nd prize in a competition and it has been published - but I don't read German well enough to spend time finding any more. http://en.wikipedia.org/wiki/Shouryya_Ray

 

[Hurkyl]

Whatever it was, it won him 2nd prize in a competition and it has been published - but I don't read German well enough to spend time finding any more. http://en.wikipedia.org/wiki/Shouryya_Ray

Read the talk page. At this moment:

The article's only citation is to an Indian news website which repeats the claims of the British tabloid The Daily Mail. This is not a reliable source. —Psychonaut (talk) 11:23, 27 May 2012 (UTC)
yeah, I'm seeing this story everywhere but Can't find any details on the actual math involved.144.132.197.230 (talk) 11:38, 27 May 2012 (UTC)
Where is the maths problem and what was his solution? 220.239.37.244 (talk) 11:44, 27 May 2012 (UTC)
ha I guess I am not the only one looking for the problem. It's just annoying when you hear something like an unsolved problem in physics and they don't tell you the actual problem. — Preceding unsigned comment added by 76.197.8.154 (talk) 12:17, 27 May 2012 (UTC)
Page should be deleted and recreated some time in the future if the story turns out to be true. It's too soon and Wikipedia is not a news source. There should also be a verifiable citation of the nature of the two problems in question and that they actually were regarded as unsolven previously. 82.6.102.118 (talk) 14:02, 27 May 2012 (UTC)​

 

[Jimmy Snyder]

The problem is to calculate the trajectory of a body thrown at an angle in the Earth's gravitational field and in a Newtonian fluid. His paper claims to be the first analytical solution to the problem.

 

[ObsessiveMathsFreak]

The problem is to calculate the trajectory of a body thrown at an angle in the Earth's gravitational field and in a Newtonian fluid.

In "Mathematical Aspects of Classical and Celestial Mechanics", Arnold & co. claim that this problem was solved by Legendre for a wide class of power law resistance terms of the form $$c v^\gamma$$. The extract is attached.

Arnold claims that the 1st order equation which the system reduces to is soluble by the method of variation of parameters, but when he says something like this you always get the impression he's ducking out. But what do you know, Wolfram alpha solves it so I assume the method must work eventually.

Maybe the solution here is for more complicated force laws, or for a particle which perhaps has angular momentum or something? Of course, it's also possible that everyone (outside of Russia) simply forgot that the solution had ever been found.

 

[CAC1001]

How do they know he figured out the actual solution if it has stumped mathematicians for so many years? That said, things like this have happened. There was a woman who, purely by random chance, figured out how to solve some kind of mathematical color theorems that had stumped mathematicians for many years.

 

[AlephZero]

Whatever it was, it won him 2nd prize in a competition and it has been published - but I don't read German well enough to spend time finding any more. http://en.wikipedia.org/wiki/Shouryya_Ray

Read the talk page...

This seems to be the primary source for the story. http://www.jufo-dresden.de/projekt/teilnehmer/matheinfo/m1 Computer translation:

Category: Mathematics/computer science
Supervisor: Prof. Dr.-ing. Jochen Fröhlich, Dr.-ing. Tobias Kempe
Type of competition: Young researchers

Prizes won:
•2nd place in the State contest
•National winner
•Regional winner for the best interdisciplinary project
Two problems of classical mechanics have withstood several centuries of mathematical effort. The first problem is therefore, to calculate the trajectory of a slanted raised body in the Earth gravity field and Newtonian flow resistance. The underlying power law was already discovered by Newton (17th century). The second problem, the goal is the description of a particle-Wall collision under Hertz'scher collision force and linear damping. The force of the collision was already in 1858 derived from Hertz, a linear damping force is known since Stokes (1850).

This work is the analytical solution of this so far only approximate or numerically solved problems so to the objectives. First the two problems in the context of generalized solved full analysis, these are then compared with numerical solutions and finally starting inferred statements about the physical behavior of the analytical solutions.

Without seeing his actual competition entry, comparing it with any previous work is just speculation IMO. Perhaps the press is ignoring the second problem because it doesn't have an nice headline like "Indian kid is smarter than Newton".

 

[K^2]

But what do you know, Wolfram alpha solves it so I assume the method must work eventually.

Alpha solves it because it interprets your equation for u[k], rather than u[a]. Partial u with respect to alpha is zero, in this case, so naturally, solution is just the remainder of your equation, which is a non-differential equation. You really should never rely on Alpha to interpret your equations correctly. Always double-check. Better yet, skip Alpha and use Mathematica.

Mathematica does solve this equation down to an integral which probably cannot be evaluated analytically.

 

[mfb]

How do they know he figured out the actual solution if it has stumped mathematicians for so many years?

Solving problems can be tricky - checking the solution is usually much easier.

While the computer translation in AlephZero's post is a bit funny, it contains everything relevant. The actual problem and the solution are not given.

 

[gturner6ppc]

I've also been searching high and low for his paper, to no avail, though I did run across one photo of him holding his equation, which looked quite simple for such a vicious problem. (The drag on a projectile is a function of the velocity squared (with caveats), and the velocity decreases based on the drag. The current method of solving the problem is iterative interpolation using data from standard reference projectiles.)

From some other references, I gather he approached it as a damping problem, mentioning attempts at it by Hertz and Stokes.

 

[russ_watters]

I'm having deja vu - this just happened a few months ago.

 

[AlephZero]

From some other references, I gather he approached it as a damping problem, mentioning attempts at it by Hertz and Stokes.

I took the computer translation as meaning they were essentially two separate problems, the second one being Hertzian contact with the wall (including some model of energy loss during the impact).

If he has achieved anything significant on the contact/impact problem, I would be professionally interested in seeing it. Modelling this numerically as part of a larger mechanical system is usually a PITA.

 

[D H]

But what do you know, Wolfram alpha solves it so I assume the method must work eventually.

It's always a good idea to check whether Mathematica made some amazingly dumb mistake. And it did.

Here is the correct solution: http://www.wolframalpha.com/input/?i=u'(a)+k*u(a)*tan(a)++k*c*arccos(a)=0. Note that the solution contains a definite integral.

What about that definite integral? As an indefinite integral, Wolfram alpha just gives up . http://www.wolframalpha.com/input/?i=integrate+(arccos(x)/cos(x)^k)*dx As a definite integral it times out.

 

[lugita15]

Did a 16 year old solve a centuries-old problem by Newton?

It seems like either something important (I'm not sure what) has just happened, or this is just another baseless tempest in a teapot manufactured by the media. But see here. They're claming that a 16-year old kid named Shouryya Ray just solved a problem posed Newton centuries ago, concerning the trajectory of a particle in the Earth's gravitational field subject to air resistance. They're also claiming that in the course of his work, he solved a problem of linear damping in a Newtonian fluid posed by Stokes in 1850 and another linear damping problem concerning collision of a ball and a wall posed by Hertz in 1858. Apparently for this work he won 2nd place in the national high school science competition in Germany.

Here's the abstract or description of his work (via Google Translate):

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 been known since Stokes (1850).

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 generalized context, they then compared with numerical solutions and, finally, on the basis of the analytical solutions derived statements about the physical behavior.

What's going on here? Can anyone find out any details about this if it's significant, or this is just a much ado about nothing?

 

[K^2]

Here is the correct solution: http://www.wolframalpha.com/input/?i=u'(a)+k*u(a)*tan(a)++k*c*arccos(a)=0. Note that the solution contains an integral equation.

What about that integral equation? Wolfram alpha gives up on that. http://www.wolframalpha.com/input/?i=integrate+(arccos(x)/cos(x)^k)*dx

That's not an integral equation. That's just an integral.

 

[jtbell]

Threads merged.

 

[D H]

That's not an integral equation. That's just an integral.

Corrected.

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

DSolve[u'[a]+k*u[a]*Tan[a] +k*c*ArcCos[a]==0, u, a]

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/article/pii/S0020746201000683

 

[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]

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!

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]

the media is blowing things way out of proportion.

I'm for this choice.

 

[Ayre]

On reddit:

http://www.reddit.com/r/math/comments/u74no/supposedly_this_is_a_new_formula_for_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/images/2012/index/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

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

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 learned to be more careful before claiming new solutions to old problem.

 

[Hurkyl]

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

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

 

[Ayre]

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

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 realize 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]

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 ..."

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.

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?

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

$$\dot u(t) + \alpha u(t) \sqrt{u(t)^2 + v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g$$

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:
Yabugarbagea, K., Yamagarbagea, 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]

I
When the paper was almost done, I discovered a 2007 paper that claims to have found an analytic solution. Here it is the reference:
Yabugarbagea, K., Yamagarbagea, 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.

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]

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:
Yabugarbagea, K., Yamagarbagea, 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.

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.

 

[Euler1707]

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

$$\dot u(t) + \alpha u(t) \sqrt{u(t)^2 + v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g$$

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.

Thank you very much for this!