# Unanswered questions in Classical Mechanics?

1. May 31, 2013

### Dmobb Jr.

Are there any unanswered questions in classical mechanics?

By unanswered I mean unanswered and attempted. I could easily think of a question which has never been asked.

Edit: Sorry about the misspelled thread title.

2. May 31, 2013

### Mute

Depending on where you draw the line with "Classical mechanics", there are numerous open problems listed on this Wikipedia article.

Probably one of the most extensively studied and yet still unsolved "classical" problems on that list is turbulence.

3. May 31, 2013

### mishima

Like what?

4. May 31, 2013

### Bobbywhy

From the referenced Wiki article I notice at least one natural phenomenon that is so far unexplained: Ball Lightning.

5. Jun 1, 2013

### physwizard

How about the 3 body problem?

@Bobbywhy : Ball lightning isn't just a mechanics problem.

6. Jun 1, 2013

### f95toli

The "problem" with the 3-body problem has nothing to do with physics: it is all math.

Writing down the correct equations for the 3-body problem is not difficult, and solving the resulting equations is easy using numerical methods. However, it turns out that there is no general closed analytical form for the problem which is why some people claim that this problem is "unsolved" (note that it is sometimes possible to write down to the solution as an infinite sum).
In my view this is a bit silly since there are very few real-world problems that can be solved analytically: just about EVERY interesting problem in physics requires numerical methods. I is just the way the world works: its complicated.

7. Jun 1, 2013

### cgk

To add to f95toli's answer: There is no fundamental difference between "elementary trigonometric functions" (including cos/sin/exp/ln, etc) and any other function you can write down in the form of an initial value ordinary differential equation. Some might be a bit easier to evaluate numerically than others, but that has no /theoretical/ consequence. All of them are well defined, and an algorithm that can evaluate them to any required precision can easily be given (and being able to approximate a solution to any required accuracy is nothing else than having solved the problem exactly). Claiming that the three-body problem is "unsolved" is ridiculous.

8. Jun 1, 2013

### physwizard

Okay, so what is the general analytic solution then if it is 'solved'? Can you write it down here either in closed form or as an infinite series? For the most general case(no simplifying assumptions, no restrictions, no approximations) ? Positions of the three masses as a function of their initial positions, initial velocities, and time ?

9. Jun 1, 2013

### f95toli

Again, why would you want an analytical solution? They are only useful if they give you some insight into the problem, OR they they are more efficient for calculations than purely numerical metods (e.g. direct ODE/PDE solvers). It turns out that for the 3-body problem analytical solutions aren't very helpful.

You can find more information about the various analytical solutions (Sundman etc) on the wiki page for th 3-body problem. But again, the solutions (when they exisit, sometimes they don't) are of virtually no practical use.

There is also some very recent work on orbits which involve the 3-body problem, where new solutions were discovered (numerical work).

See
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.110.114301

10. Jun 2, 2013

### guest1234

A very possible explanation: magnetophosphenes

11. Jun 2, 2013

### Bobbywhy

Will you please say why this proposed explanation is "very possible" as compared to all the other proposed explanations? Thank you.

12. Jun 4, 2013

There are a lot of unanswered questions in classical mechanics. Unfortunately I am no expert and I just have a vague feeling about the type of topics where there is definitely work to do.

Arbitrary rotating bodies with forces. There seems to be only a horrendously complicated solution for the force free case.

Periodic solutions of the three body problem. Can they be classified?
http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whopping.html [Broken]

You can reach infinite speed within finite time in classical gravitational problems. There are explicit solutions, but can the solutions be classified?
http://plus.maths.org/content/outer-space-twos-company-threes-crowd

Chaotic systems. What can we say about them. What Lyapunov exponents can we calculate? Can we classify these systems more/better?

What systems can be proven to be ergodic? How rigorous can one prove the laws of thermodynamics for classical systems.

Variational calculus seems to need some work. Lagrange mechanics. Hamilton-Jacobi Formalism applied to something else than the harmonic oscillator.

Apparently a polyhedra that is also a Gömböc exists http://en.wikipedia.org/wiki/Gömböc construct one with the minimum number of faces.

All that stuff is just rigid body mechanics. Continuum mechanics is a whole field of its own where we still know very little especially about turbulence.

Last edited by a moderator: May 6, 2017
13. Jun 4, 2013

### Rap

It depends on what you mean by "unanswered questions". Does classical mechanics provide equations that are solvable in principle for every phenomenon within its range of applicability? (c/v->0, $\hbar/A->0$ where A is the action). The answer is yes.

Are there equations which cannot be solved using present mathematical/computational techniques? The answer is yes. If mathematical/computational techniques advance at a finite rate will there still be equations that cannot be solved a finite time from now (like 10 gazillion years from now)? Again, yes.

14. Jun 4, 2013

### retro10x

How often are new questions asked in the first place? In other words, how often is a new phenomenon observed that people think can be explained classically? Does this even happen anymore?

15. Jun 5, 2013

16. Jun 5, 2013

### Dmobb Jr.

Sorry I have ignored this thread that I started.

These ansers are not quite what I was looking for. The "problem" does not have to be an unexplained physical phenominon. I will explain what I mean with an example. Last year my friend asked his physics proffessor, "If you have two rods spinning about there center with mass m and angular velocity omega and the end of one rod attaches to the end of the other, what will be the resulting angular velocity?"

The proffessor was able to solve it (although it was quite hard) but he said that he had never seen a problem like this before.

Basically I am looking for simple mechanics problem that are really really hard.
The physics equivilent of fermats last theorem (looks easy but is not).

17. Jun 5, 2013

### Rap

The problem of the rods that you called hard was hard because it was hard to find an analytic solution. There are loads of such problems, I'm sure.

Most problems are not analytically soluble, you have to do a numeric solution. Numerical solutions to chaotic systems (three body, double pendulum, etc.) can be numerically unsolvable in a practical sense past a small time interval, requiring computers larger than the size of the known universe, etc.

Fermat's last theorem is an unresolved theorem in number theory, so I guess to ask a question about any physical system that requires an answer to such an unresolved theorem would qualify. I don't know of any such thing in classical physics, but I bet they exist. In quantum systems its probably much easier, since you tend to deal more in integers (quanta) rather than continua. For example, in the theory of the Bose-Einstein condensate, the energy of a state is characterized by the sum of the squares of three integers. When that sum is unique there is only one such energy level, but when you can have a number of different sets of three integers giving the same energy, then there is degeneracy, and you have to treat things a bit differently. If I remember correctly, the statistics of such a situation can get really sticky, unless you make some approximations that hold for large numbers, but then you can't deal with the small number situation, so then you make some small number approximations, and are left with an intermediate range that is pretty hairy. But thats not classical mechanics, that's statistical mechanics, so cancel that.

18. Jun 5, 2013

### micromass

Staff Emeritus
It has been proven over 15 years ago by Andrew Wiles.

19. Jun 5, 2013

### Rap

Oops. Right.

20. Jun 8, 2013

### lightarrow

Classical mechanics includes even General Relativity. In GR it's easier to write the questions which have found an answer than the opposite...