Three body problem

[abskebabs]

I remember reading about the three body problem from the advance notice article I received for my OCR synoptic physics exam. In fact, I have it here with me now.

 

To get to the point, it interested me very much and made me wonder if it could have been solved, and worry; rather stupidly that it might be on the exam. Anyway, I'm glad it wasn't, as the article stated there is no comlete solution to the problem. Why is this? Are the solutions to this problem only approximate? If so, are methods involving matrices involved?

 

Also the french mathematician Jospeph Louis Lagrange is credited with finding special places where the smallest body of a system(consisting of an asteroid, the sun and the earth as an example) consisting of three, could keep the same position relative to the other two.

 

It seems to me, it would be relatively easy perhaps working out where these "special" places are if variables of the 2 larger bodies, like the masses, velocities and orbital radii are known.

 

It is great that even classical or basic physics can be so interesting:-) .

[Royston]

It is great that even classical or basic physics can be so interesting:-) .

 

The three body problem came up in my QM section of my current course, the problem is that it isn't that basic at all. With two bodies classical mechanics can be applied, but as soon as you include a third body (like your example) an asteroid, then even though the three masses are in the same plane, the effect of the second largest body should have no effect on the asteroid i.e the effect has to be so neglible that it doesn't alter the orbit.

 

IIRC they are looking to QM to solve the problem (but please correct me if I'm wrong on this.) I know they've used hamiltonian systems and something called a Jacobi integral (might have the name wrong) but if you think of it in the terms of gravity effecting a particle - where obviously the effect is incredibly small, I think this is why they are turning to QM for answers.

[abskebabs]

Holy moly:eek: ! That's news to me. I had no idea that this problem would require quantum mechanics, I just thought statistical methods could be applied to produce approximate solutions to this problem. For example, I thught perhaps one could apply nonlinear equatons and solve them using matrices to produce approximate answers.

 

I hope someone else can eleborate a little more on this, as this has now left me proverbially scratching my head:confused: .

[timo]

had no idea that this problem would require quantum mechanics, ...

Me neither. And given that the problem is analytically solving the equations of motion (which is solely math and no physics at all), I have serious doubts about it.

 

I just thought statistical methods could be applied to produce approximate solutions to this problem. For example, I thught perhaps one could apply nonlinear equatons and solve them using matrices to produce approximate answers.

I don´t understand that. But you can get approximate solutions by simply simulating the system numerically (on a computer).

[Royston]

My bad, it was only brought up in the course as it was a good example of an unsolved problem in physics...I know there is a three body and many body problem on particle size scales, but that's a different kettle of fish altogether.

[abskebabs]

I don´t understand that. But you can get approximate solutions by simply simulating the system numerically (on a computer).

Thinking about it now, I think I was talking a load of garble and u need not worry about it. It's just I've recently been reading up about matrices and how they can be adapted to solve systems of equations by producing an augmented matrix; and then performing elementary row operations, to obtain a triangular matrix. I thought something similiar could be done here with gravitational equations.

 

I used the word statistical, and that has no bearing on what I was trying to say whatsoever. I know that it is easy to find exact solutions to linear systems of equations, but only approximate solutions can be achieved whwn trying to solve nonlinear systems of equations. I then made the link between this and the inabillity to solve this specific problem exactly.

 

Was I fallacious in making these assumptions? Also, on a side note what kind of application of matrices are used in Heisenberg's matrix mechanics?