Op zondag 10 maart 2002 om 18:41:45 UTC+1 schreef Chris Hillman:

I wrote:

Transfer operators (infinite dimensional generalizations of "transfer matrices") and zeta functions (generalizations of the Riemann zeta function) were defined in a general dynamical setting by Ruelle, and
they have been intensively studied by Mark Pollicott and Viviane
Baladi, among others.

[snip]
Nicolaas Vroom replied:

This subject is part of ergodic theory, which evolved directly from the studies of Poincare on solar system dynamics and the proposals of Boltzmann concerning statistical dynamics (and the objections of Zermelo and others to Boltzmann's proposals).

This is an excellent post

However I am still trying to find an answer on the following three questions:
1a What is the state of our Solar system over 100 million years ?
1b Can we predict this state ? (Accurate ?)
2. Is our solar system stable ?
3. Can we use the chaos theory to answer the first two questions ?

Thanks for the praise, but I thought I had made it clear that, first, (2)
is really many questions, because (a) there is more than one good notion
of stability, and (b) there is more than one relevant dynamical time
scale, and second, that the theory of dynamical systems not only greatly clarifies these distinctions, but gives answers to questions of stability which yield valuable insight.

Recently I found this posting again.
The most important question is #2.
The correct way to answer that question (and similar questions) is by performing observations over a long period. How longer the better.
A practical example is by observing a star cluster.
What we should observe is that the positions of the stars will change
and if you are lucky that two stars will collide.
When that is the case we can define that the cluster was not stable.
What is also possible that a star will escape from the cluster.
Also in that case the cluster was not stable. The whole point is the
only way to answer the question is by performing observations.

A different way is to use the theory of dynamical systems.
A good overview is https://en.wikipedia.org/wiki/Dynamical_systems_theory. "Dynamical systems theory is an area of mathematics used to describe
the behavior of complex dynamical systems, usually by employing differential equations or difference equations." There are 11 related fields.
This raises a new question: does this theory answer question #2?
To be more specific: can the question be answered by means of mathematics?
Of course there are systems which can be described by means of mathematics
but this are all mathematical systems, not physical systems.

Is it possible to decide by means of mathematics if a roulette is correct?
No. The only way is by performing an experiment 1000 times and based on the results to define the roulette as 'correct', otherwise certain corrections
have to be performed.
And what about our solar system. Original this was also a cluster of let me
say 1000 small objects. During the ages the stars collided and merged. Slowly the number of objects became smaller and a large one appeared at the centre
and smaller ones started to revolve around this large one. At least that is
the way we think it physical happened.
To test this we can try to simulate this system on a PC and use for example newton's law. We start with a system with 1000 identical objects, give each
an initial position and speed and observe how the simulation evolves.
Most probably all the points will collide at one point.
To make the simulation more realistic you have to adapt the possitions and velocities such that the simulation becomes more stable. That means you
have found a mathematical solution to a physical question. But is that in agreement which what physical happened, millions of years a go?
Most probably not.

This leads me more or less to my final conclusion.
It does not make sense to study the evolution of our solar system from
a mathematical point of view starting first with 'acurate' observations
roughly 500 years ago, because our solar system is much older.
The point is that starting from that day our solar system, physical evolved more or less by itself, based on the influence by a group of local objects,
not under the influence of any mathematical law.

From a physical point of view studying this evolution there is no accuracy issue.

The same can be said that there is no chaotic issue.

From a mathematical point of view there is an accuracy problem.

They come to light whan you perform a mathematical simulation of your system. May be most important are the measurements of positions of the objects involved and the time of these observations. Using these observations the speed and the accelerations of the objects involved are calculated.
A second accuracy problem happens when objects come near each other. This is also
called the chaos problem. To decrease the step size can be a solution.
The third problem is the mathematics or equations used as part of the simulation,
specific all the parameters involved.
If the equations use the speed of light also that parameter has to be calculated,
based on observations. You can also call that the fourth problem.
For the problems involved, and the impossiblity, select this link: http://users.pandora.be/nicvroom/wik_Dynamical_systems_theory.htm#ref2

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