pariah

edit: just saw bagass post. nm.

NialScorva

It's only simple if one pendulum is small enough relative to the other that you can consider it to be zero. Otherwise it's pretty a pretty hairy problem.

Quote:

So what systems can we NOT guess the outcome of yet, that would benefit us to be able to? And, an astronomy question, how is our solar system chaotic? I could understand the universe being chaotic in expansion (another area I don't understand well), but...

The solar system is a many-body system. Every planet exerts gravity in some small way on every other planet. You can come close by assuming that this pull is zero except in very large or very close objects to your area of interest.

Many body problems are very hard, the vast majority of the time you cannot solve them at all. At best you can get a good approximation.

tensorproduct

The most obvious one is the the weather. People talk about butterflies and hurricanes, but with more accurate theories and more computing power we can be more certain of where a tornado won't appear out of nowhere.

Coragyps

Adding to what's already been said: chaotic behavior is fairly easy to see in the asteroid belt, where you have not just a "many-body problem" but a "buttload of bodies problem." Asteroids are frequently bumped into new orbits in a chaotic fashion - that's how things the the end-Cretaceous dinosaur-killer get here. There are lots of studies on just how these orbits evolve, and they explicitly tie chaos into their simulations.

Shadowy Man

Nice! Especially for someone who has had the painful experience of working through the Hamiltonian for the double pendulum system. For a prelim exam, no less!!


As far as the solar system being chaotic. Isn't any N-body central force system chaotic, as long as N > 2 ??

Lobstrosity

Hmm, working out the Lagrangian for a double pendulum isn't all that difficult, and though it's been a little while, I don't remember it being very difficult to obtain the Hamiltonian from the Lagrangian. The big pain is trying to solve the equations of motion one obtains from the Lagrangian (mainly because you can't).

Also, I don't think all three-body problems are chaotic. For example, there exist Lagrange points around two orbiting masses at which another mass can reside without (I think) inducing chaos, though I could be very very wrong about this.

Shadowy Man

Well, L1, L2, and L3 are unstable equilibrium points, but L4 and L5 are stable (depending on the mass ratio of the two primary objects). However, they are most likely stilll chaotic, i.e. non-analytic solution to the motion, highly sensitive to perturbations.

tensorproduct

Not necessarily, it's been a while since I did a course on Chaos, but there are certain conditions where the system won't be chaotic. If you take a three body problem with S being the most massive and E and J orbitting S, how "chaotic" the system is depends on the ratio of the orbital times of J and E. And the system is least chaotic when the ration of orbits equals phi*.
Like I said it's been a while so I could just be talking outta my ass.


*phi = [sqrt(5) + 1]/2 "Golden Ratio"

Santas little helper

If the 3-body or the "much more than 3"-body problem were always chaotic then the earth ( and every other planet ) would have a
totally crazy orbit so we probably wouldn't be here to talk.
For the 3-body problem if for example the 3 point masses occupy the corners of an equilateral triangle then the circular orbits are stable.
Unstable equilibrium points is
a necessary but not sufficient condition for
chaos.I don't think there is a
formal definition of chaos.So for example I don't
think that there is some mathematical definition
which says if a system of differential equations
is chaotic or not.
Chaos is characterized by orbits
which are dense in some region of space which means that for every point A in the region and
every real d>0 there is a point B of the orbit such that the distance between A and B is less than d.
Chaos is also characterized by patterns repeating
in smaller scales ie if you magnify some region of
the orbits you see similar patterns in the magnified region to what you saw in the original region like the Mandelbrot set.

My limited knowledge of the subject comes mainly
from a module I took many years ago so some of the
above may be less than accurate.

As for literature I believe that the classic introduction for the layperson is supposed to be
"Chaos.Making a new science" by James Glieck.But
I haven't read it myself.You can find a review for
it here

Finally I have a question myself:How is it possible to simulate the behaviour of a continuous
chaotic system using a computer ? I would imagine
that the very description of "chaotic" makes this
impossible.Perhaps what the simulations linked above do is create a new chaotic system.
What I mean is that if you take some system of
differential equations which describe a chaotic
system and use some of the numerical methods available to solve it then the orbits you get on your computer
screen will have nothing to do with any orbit you'd get if you actually "run" the system in the
real world.Nevertheless it seems likely that you
do create a new discrete chaotic system in your
computer.

PaulPritchard

I read this a few years ago and would strongly recommend it. There are some amazing ideas in there, all of which are very well explained.