Many Body Problem: Physically & Mathematically Explained

[Ed Quanta]

Using Newtonian physics, I have heard that it is impossible to define an equation or set of equations that describe the motion of three bodies interacting under the force of gravity. Can anyone demonstrate this physically or mathematically? I don't see how the equations become unsolvable.

 

[Integral]

You certainly can define equations which describe many body problems. But they do not have a closed from solution. They must be solved numerically and since they are describing a non linear dynamical system the solution is falls into chaotic behavior.

 

[Ed Quanta]

I apologize for using the wrong word "define". I guess what I am curious about is how the equation becomes nonlinear or rather does not have a closed form solution.

 

[Integral]

Some Googling turned up this page which seems to have some good infromation on the problem, along with some nice java applets (look under current projects).

 

[ZapperZ]

Using Newtonian physics, I have heard that it is impossible to define an equation or set of equations that describe the motion of three bodies interacting under the force of gravity. Can anyone demonstrate this physically or mathematically? I don't see how the equations become unsolvable.

To make sure you don't get into a state of confusion, note that your question actually is more related to N-body problem. "Many-Body" problem typically is reserved for a "gazillion" body interaction where many-body techniques are used, especially in condensed matter physics (See, for example, G.D. Mahan's standard text "Many-Particle Physics").

So if you are googling, it is good to make sure one knows what terminology is appropriate for what one is looking for.

Zz.

 

[dextercioby]

I'm sure that at your library u can find at least one book on celestial mechanics (describing gravitational interactions with Newtonian theory).All books on celestial mechanics should exhaust the subject of 3 body-problem.

Oh,and one more thing,add to what Integral has said.Lagrange and Euler have found particular solutions to this problem (3-body interaction in Newtonian gravity). (check "Lagrange points"...).

As for terminology,"Many Body Problem" refers to what Zapper said...

Daniel.

 

[HallsofIvy]

"how the equation becomes nonlinear "

It doesn't "become" nonlinear because there are more than 2 bodies, it is already non linear because one of the dependent variables is r, the distance between the two bodies and the force (so second derivative of r) depends on 1/r².

As for why it does not have a closed form solution: Almost ALL nonlinear differential equations do not have closed form solutions! One exception is the "one-body" problem where you assume one of the gravitating bodies is so massive compared with the other than it can be considered and unmoving source for the force. The two-body problem can be reduce to "one-body" by using the center of mass and treating each body separately.

 

[Ed Quanta]

Thanks again, you guys are a lot better than some of my professors.