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I'm looking for simple problems in theoretical mechanics that are impossible or unreasonably difficult to solve by means of "brute-force" numerical integration of Newton or Euler-lagrange equations.

I'm interested in these beacuse I noticed that kind of "computer-nihilism" point of view is getting popular (at least among some students): A person says, that "in the end we anyway doing real stuff by computer simulation. And for the numerical values of parameters we are usually able to numerically obtain the result with a given precision. So we just need to know how to write down the equations".
And, therefore, "there is no need to learn all that complicated stuff in theoretical mechanics".

Apart from obvious counter-arguments for this, I'd like to show that there are basic problems you are unable to solve without "the complicated stuff".
Let me give an example of such a problem:

  1. A center, that creates some strange field with the potential $U(r)=-\frac{\alpha}{r^3}$. (Mysterious planet)
  2. A body with mass $m$ scattering off this center. (Our space ship.)
  3. A radius R, at which we want to stay as long as possible.

Find: the impact parameter $\rho$ and the energy $E_0$ for our body, so it will stay in the "ring" $R<r<2R$ for as long as possible.

The problem is easily formulated. And it is easy to solve even for "newbies" in theoretical mechanics. The specific feature of the problem -- there is no reasonable way of solving it by doing straightforward computer simulation.

Can you propose other examples of problems with these properties?

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I'm making this community wiki, since it's a "list" question. – David Z Jan 25 '11 at 18:01
Maybe somewhat orthogonal to your question but if you're interested in whether or not orbits are precisely commensurate, to approximate amounts to resigning yourself to failure! Fat load of good confining a value to a small interval does when you're trying to prove or disprove rationality! – Vandermonde Oct 24 at 8:14

5 Answers 5

The SIAM 100-Digit Challenge springs to mind. Problem 10 gives the flavor of the type of problems put forward in this challenge:

A particle at the center of a 10 x 1 rectangle diffuses until it hits the boundary. What is the probability that it hits at one of the short ends (rather than at one of the long sides)?

The answer needs to be accurate to at least 10 digits. And no, straightforward Monte Carlo is not going to help you.

And... there is an exact solution...

This is not a strict mechanical problem, but it brings across the message that certain physics problems are best solved analytically. If you insist on a mechanical problem, you should look at problem 2 which is basically a billiard problem.

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I think this is just a question of defining what you mean by "solve". All physical problems, most definitely so in classical mechanics, can be posed as differential equations for which solutions (i.e. trajectories of the dependent variables) can be found at least through numerical integration. In this sense, as far as I know, no one has identified any non-computable phenomena in physics. *

This does not however mean that all quantities of interest (which aren't explicit dependent variables in your differential equations) can simply pop-out of a computer program. Computers can only give you answers for questions that you define perfectly and precisely - and the best such example is a PDE for which you want numerical solutions.

A computer can give you precise values for the impact parameter and energy that you have asked for in your question, but in order for it to do that, you will have to define to the computer what impact parameter and energy are, and in trying to do so you will have solved your mechanics problem :)

To summarize, it is easy to ask a computer for trajectories, since it can calculate these from the relevant (P)DEs. But to get other quantities of interest such as energy or impact parameter, you will have to extract them from the physics of the problem and translate it into a form the computer will understand, which essentially involves doing some more work on paper.

* (Is there a better way to leave a footnote?) Of course, you probably already know that most problems are not analytically solvable (classic example is the full 3-body problem) and also that the most interesting numerical problems are simply intractable - the prominent example being the quantum simulation of many-body systems in condensed matter physics.

Also, if it turns out that some new physics that is discovered is indeed non-computable, there are those who claim that it might be relevant to the problem of consciousness (See Penrose's "Shadows of the Mind" - and I'll leave you to be the judge of that)

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Thank you. I do realise everything that you've said. And while I'm not totally agree with you -- you still have missed the point. I'm asking for the simple concrete examples of problems. – Kostya Jan 25 '11 at 16:53
Kostya, my point is that any quantity that is not an explicit dependent variable in the equations of motion for your system are indeed examples of things that will not be automatically churned out by your simulation. So there are plenty of examples that fit the bill of what you were thinking. But by properly defining these quantities in your program, they too can still be calculated. Nothing in physics is in principle non-computable. – dbrane Jan 25 '11 at 17:07
Can you present a working program that gives the solution to my example problem? Sometimes when one says that he knows how to do something "in principle" means that he doesn't know how to do it. – Kostya Jan 25 '11 at 18:10
No, I'm not even going to try. Because what I've been trying to convey in my answer is that the only way it makes sense to pose your question to a computer is in the form of a differential equation or an integral (or a simple algebraic result), and that for this particular example, once you have reached that stage you will already call it a solution. – dbrane Jan 25 '11 at 21:19
I'm increasingly getting the feeling from your comments and those of kakemonsteret's that you need to rephrase your question. You aren't actually looking for questions for which "there is no reasonable way of solving it by doing straightforward computer simulation" but you're looking for problems for which an iterative numerical computation is not applicable. And like I said before, yes, there are plenty of examples for which an iterative numerical computation isn't applicable - generally speaking, for quantities that aren't explicit dependent variables in your differential equations. – dbrane Jan 25 '11 at 21:23

I think you can construct such an example by considering the large scale properties or evolution of some chaotic system, like a gas, you would have a problem (time-wise) if you were to track each particle in CM, but thermodynamic formulas gives instant answers.

Edit: This one: Given two point-particles, will they ever impact? GL simulating and checking r(t)=0. Whereas angular momentum consideration gives instant answer.

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Yes. I thought of that -- Liuville's theorem for example. But this is statistical physics -- too complex for "newbies". – Kostya Jan 25 '11 at 16:58
@kakemonsteret: Not true, you can still write an algorithm that will give you a yes-no answer as to whether the body will break from orbit, if you precisely specify all initial conditions. – dbrane Jan 25 '11 at 17:13
@dbrane Without simulating the evolution? How, without using energy? – user1708 Jan 25 '11 at 17:14
Why simulate the evolution? A simulation is a solution for only the position and momentum. So it's trivially true that a pure simulation can only answer the question "what is the position and momentum at time t?". But just as you can calculate the instantaneous energy or angular momentum by adding lines to your algorithm (easily done for E and J), you can modify your algorithm to answer all meaningful questions about the system. Then it ceases to be a simulation and becomes a specialized algorithm to answer your specific question. – dbrane Jan 25 '11 at 17:19
And in most cases, the specialized algorithm won't have to numerically integrate but just use information that you have analytically derived - such as the inequality for energy that tells you whether you have a bound orbit. – dbrane Jan 25 '11 at 17:21

Not sure if this has been mentioned but the classic N-Body problem is computationally intractable. Hence all the statistical methods when N is somewhat large. This is because the problem is effectively O(N!) since every body must interact with every other body.

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I remember during my PhD I had a discussion with my colleague about mechanics problems that were still unsolved. He mentioned that it was not proven mathematically that a coin falling on the floor will, while settling down, speed up and produce that characteristic "ringing sound". I don't know what the current status of that problem is and how to look for it, since I don't know the name of the problem, if it has one. But it sounds like something interesting. It seems the difficulty in solving this problem is to see how far one can simplify it without throwing out what is essential for explaining the phenomenon.

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