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Author Topic: Dynamical systems: celestial mechanics, Poincare, Laplace, butterfly effect  (Read 2564 times)
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« on: November 08, 2007, 09:14:05 PM »

There is a work in history of astronomy I've been preparing for the most of the time in living memory. I can't say is it late, or due, or long, but counting from the 3 books I've seen, the one in focus of some private interest is Michael Hoskin "Cambridge Illustrated History of Astronomy".

On page 196 he is talking about Newton's mechanics in 20th century. He claims for example that:

1. Newton's celestial mechanics is still fine for orbital issues of most planets and satellites
2. measuring time is not and is fully done by relativistic mechanics
3. precision of measurements and precision of calculations cloud an ideal image of perfect knowledge ("Laplace demon")

Poincare pointed out in 1890s that small changes in initial conditions or orbital parameters create drastic differences over time. "Initial uncertainties for Earth increase by a factor of 3 every 5 million years, so that an initial error of 10 meters produces an error of 1 million kilometers after 100 million years".

The idea of predictability (once was confirmations for Newton's theory) is lost today as we know now for "dynamical chaos".

Previous attempts to prove solar system stability are "fundamentally flawed". Systems subject to dynamical chaos are sensitive to chance perturbations. These are the "limitations" to what can be computed, but knowledge of dynamical chaos offers implications to future models.


What struck me in the least is not that statement alone. The whole story is already too well known in form of a new kind of math, art, era of computers and artifacts.

To give a slight more dose to the background, I must add that in "Motion Mountain" Physics Textbook by Schiller it is said that butterfly effect does not really exist because of the friction and dissipation. Of course the field of observation was on some basic level of mechanics. Thats not such a big deal if we formulate that you can not expect a brief and limited event will have a grandiose outcome remotely in the future time with a great level of detail, consistency etc.

When I take that statement back into Hoskin's words I find that he was writing of the uncertainties as if they were alive somehow - or better said, active components to solar system as a whole.

By now this post has gone into multiple areas of interest. It doesn't have to be solar system, it can be any mechanical system with it's own dynamics.

Schiller's statement is really among the most sobering ones but he didn't expand on it. If we want to hire a butterfly to create a hurricane there is a bad prospect to that. If we take a look at everything else recorded and studied so far there is no phenomenon that would preserve initial action and carry it to a great distance or towards greater magnitude like that.

I expect that Schiller and me be corrected here with any information you may have.

To further the thought, my idea is to compare the problem of solar system with the problem of butterfly effect.

Newton for example believed at first that orbits are a result of attractive and repulsive forces that position the planets precisely where they are. Under the influence of Robert Hook probably, he realized that straight unaffected motions are bent into curves under one attractive force. The old astronomy and cosmology ended with that thought and new era started till this day possibly.

By comparison the inability to predict the long term state of solar system reminds me of butterfly effect in a sense that margins for errors propagate over time and space until they become "hurricanes". Thats so very odd to me - at least today.

That bring into question ideas of physical law, and principle of nature. The states of planetary bodies or parameters that are subject to chance and randomness appear to belong to the same field of free motion, or freely occupied states, degrees of freedom. In other words, there are both times and surfaces along the orbit to pick up perturbations, while not looking at the pull of gravity for the moment. The process of magnifying initial errors coincidently resembles a kind of "friction and dissipation".

The change in my view consists in that we first had an image of "errors" that are defined depending from the procedure of taking measurements and putting them into computer, and now we're slightly closer to that notion in relation to basic principles of nature, the like is thermodynamics (eg. dissipation).

Hoskin said that first attempts to prove stability of solar system were fundamentally flawed. Therefore my question is in the least, do discharges of energy as elements of thermodynamics (and influences "by chance") result in a steady pile up of uncertainty? or specific order kind of like we might want to call butterfly effect?

The interactions ought to be hidden in the fields that are subjected by higher percent of "uncertainty" and random influences. If interaction lurks in those fields then the theory of chaos (of which I know very little) would be a physical phenomenon. Otherwise it would not be physical but purely observational artifact or an error in the process of calculation.

The difference makes me feel like Newton wondering are there only attractive forces or attractive and repulsive. Thermodynamics is about flow of energy so I'd vote there are interactions at all times but I can't seem to include the sensitivity to initial conditions and such into the view as being something physical as would have been an repulsive gravity - it only it were true.

The benefit for doing this query is important as far as getting a picture of micro and macro scale phenomena. Orbits can be one example. What about modeling whole wide universe that blew up and passed apparently more than once through Boltzmann equations and other statistics? It should be big considering that those models maybe have no reality at all.

Thanks.

I also posted this question here, http://www.physicsforums.com/showthread.php?t=196680
so its a little bit of exaggeration by now. :-))
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