Can Nonlinear Trajectories Lead to Linear Equilibrium?

[Loren Booda]

Nonlinear mapping tends to maintain nonlinearity the farther its trajectory from a fixed point. Is there an example of a trajectory reverting to linearity after traversing a nonlinear region? I am especially interested in local, linear special relativity transitioning to global, nonlinear general relativity, and a possible converse.

 

[Tide]

Yes. Any object entering the Solar System from outside came from a linear trajectory, experiences a hyperbolic trajectory in the vicinity of the Sun and reverts to linear after it has left the Solar System.

 

[Loren Booda]

Tide,

Does general relativity behave nonlinearly, yet not chaotically? I believe what I seek is a near-linear gravitational trajectory becoming a more chaotic one, then becoming near-linear again. Sorry for the confusion.

 

[Tide]

I'm still not quite sure what you're looking for but here are a couple of ideas that I think resemble what you seem to be asking.

Some aspects of weather go from calm (normal weather) to chaotic behavior (hurricanes) and back again to calm.

Plasma waves can produce an "echo" meaning that when such a wave is launched it is well structured but as it propagates away it is damped (Landau damping). The wave disappears but later downstream the wave re-emerges. This process can repeat several times.

Regarding general relativity, the motion of objects (including light) close to a massive object can be chaotic but I am not aware of a situation where an object that enters close to the object (event horizon, e.g.) completely escapes.

 

[Loren Booda]

Tide et al,

To repeat: from what I know, gravitation propagates along trajectories that are generally nonlinear yet nonchaotic. Is this right?

 

[meteor]

Does general relativity behave nonlinearly, yet not chaotically?

Standard chaos theory seems not to work in GR, at least is what this paper claims
http://arxiv.org/abs/gr-qc/9612017
so they are trying to develop a new definition of chaos useful in GR

 

[Loren Booda]

Thanks, meteor. That's more toward what I was looking for.

Anyone want to define diffeomorphism invariance? All I know is that it is some kind of symmetry followed by GR.

 

[Gokul43201]

http://mathworld.wolfram.com/Diffeomorphism.html
http://relativity.livingreviews.org/Articles/lrr-1998-1/node11.html

 

[MathematicalPhysicist]

Standard chaos theory seems not to work in GR, at least is what this paper claims
http://arxiv.org/abs/gr-qc/9612017
so they are trying to develop a new definition of chaos useful in GR

how their work differ from nottale's work on scale relativity?

nottale's webpage:
http://www.chez.com/etlefevre/rechell/ukrechel.htm

 

[Phantomas]

Hello all.
Just a few observations.
Our solar sisem is chaotic by nature (large time scail). Our life span is just too short for us to notice.
Also weather is chaotic by nature, that means that in calm weather and in mid huricane it is impossible to predict future events for distant future, even with lots of acure data (not equaly unpredictable, but still unpredistable).

@LorenBoda, So you are looking for transitional chaos in damped systems. I think if you want to include general relativity you'll have to be more specific, but there is an example of a transitional chaos involving gravity.

It is an experiment with three magnets (positioned at the corners of a tiangle) and an iron ball that hangs above them exactly at the center.

When you release the ball its movement is linear (it's a damped pendulum). When the amplitude weakens it becomes chaotic (it can't decide which magnet attracts it more). When amplutide weakens even more, it settles and osscilates arround one of the magnets (yt the beginning you can't tell which one).

I actualy did the experiment once, here is a picture:

Cheers, Phantomas

 

[Loren Booda]

Phantomas,

What would be the pattern of motion if the magnet system were superconducting?

Gokul43201 and loop quantum gravity,

I will indulge in these sites you offer - thanks.

 

[meteor]

LQG: The main difference that I see is that Nottale's approach indeed modifies General relativity, while Schleich and Witt are only giving a new definition of chaos, maintaining the habitual GR structure. For example, you must know that in GR spacetime is a differentiable manifold. But looking this Nottale's paper
http://arxiv.org/abs/hep-th/0307093
he says there that in his theory spacetime is a non-differentiable manifold

 

[Gokul43201]

Phantomas,

What would be the pattern of motion if the magnet system were superconducting?

The same.

This is a classical system, no matter whether the magnets are permanent magnet bars or superconducting electromagnets.

Why do you ask ?

 

[Loren Booda]

It reminded me of a superconducting magnet that has the ability to be stably suspended without supports where a classical magnet cannot.

 

[Gokul43201]

Loren, you are a recollecting incorrectly (boy, is that a tongue-twister !). What you are talking about, is a superconductor suspended over a regular magnet.

And remember that ferromagnetism itself is a Quantum Mechanical effect, so the only "classical magnet" is an electromagnet.

A superconducting magnet is merely an electromagnet that uses a superconducting wire for the current (so you can have much higher currents, and hence much higher field strengths).

 

[Phantomas]

Exactly.
It doesn't matter what kind of magnets you use.
If they are stronger you have to use a heavier ball to get the same results, or just hang it higher.

P.

 

[Loren Booda]

Gokul43201,

What is an example of a magnetically, otherwise freely suspended object at room temperature?

 

[Gokul43201]

I can describe something that'll do this :

Take a strong (like an NdFeB or SmCo5) magnet in the shape of say a cylindrical disc of diameter D, magnetized along the direction of the axis. Set this on your table. Now over this slide a transparent (so it's nicer to see) plastic tube/pipe of inner diameter just barely greater than D, but have it be longer than the height (or thickness) or the magnet. Now take another magnet, identical to the first and drop it down the pipe. If you dropped it with opposite poles facing each other, this second magnet will levitate above the fist one.

 

[Loren Booda]

But can the levitated object be free of mechanical contact ("otherwise freely suspended")?

 

[Gokul43201]

I can't easily picture a geometry between only ferromagnets that will give rise to a stable equilibrium - but maybe I'm just not thinking right. In the above example, there is a point of equilibrium (where the magnet can levitate without any mechanical contact whatsoever), but it is unstable - if slightly disturbed, the magnet will flip and fall into the other one.

In the case of a reasonable diamagnet, it is actually possible to achieve levitation at the highest achievable fields today. With a reasonably high field magnet, it is possible to stably levitate say, a drop of water or a piece of bismuth - though I think that the better approach is to levitate the magnet between bismuth plates.

Someone actually levitated a frog with a pulsed field magnet. A google search might prove fruitful.