Chaos & Unsolvability: Solving the 3-Body Problem & the Role of Chaotic Motion

[Ed Quanta]

Can something like the 3 body problem be shown to be unsolvable because it exhibits chaotic motion? In other words, must all solvable equations of motion be 100 percent free of chaos? Sorry if my question isn't clear.

 

[polyb]

The equations of motion that describe the 3-body problem are not exactly solvable concisely because they are non-linear. Usually the best method for solving them is numerical.

Chaos is best described as sensitivity to initial conditions. Thus in a system of non-linear equations if we alter the initial conditions by any amount we will see a different behavior in that system.

Here is the wikipedia write up on chaos theory:
http://en.wikipedia.org/wiki/Chaos_theory

And one from math world:
http://mathworld.wolfram.com/Chaos.html

These should help you clarify some of the points you will need to understand if you would like to study this subject. Good Luck!

 

[Crosson]

Ed, it is possible to have chaotic motion for which the dynamic equations have a solution (for example, a driven pendulum).

 

[krab]

Ed: I think the answer to both questions is Yes, provide you are using "solvable" to mean integrable. Of course many simple systems exhibit chaos, but are still numerically solvable to some precision. But they are not analytically solvable.

 

[Antiphon]

The classical three body problem is linear because F=ma is linear. Electrodynamics
is also linear and the charge distribution on a piece of metal is a constrained
many-body linear problem. But it too is not solvable for the point particles- the
solutions are NUMERICALLY unstable.

It is chaotic because the equations display a critical sensitivity to initial conditions,
like solving for which way a pencil will fall when stood perfectly on its point.