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Ludwik!
I was (am) also highly suspicious of that numerical simulation. I
suggest you try it w/ your known example the conical pendulum.
Incidentally, not; single body orbital stability simulation has already
been discussed at an introductory level early in the history of
numerical instruction. vide: Eisberg and Lerner (1981) section 11-7
perturbations and orbit stability pp. 485, ff. (~ 10 pp.) This is the
last section of the chapter Gravitation and Central Force Motion. The
authors also discuss other than inverse square orbits. Feynman et al.
uses leap frog simulation, but I'm not going to check if he perturbed
orbits. Also a number of advanced (intermediate?) texts do numerical
including Runge-Kutta methods. I've only read them for oscillators.
bc also into numerical simulations.
p.s. I found Cooper and Pellegrini very useful as they partially solved
the coupled oscillator problem with which I'm experimenting. I just
found they analytically show that orbits w/ 1/r^n, n<3 are stable *.
pp. 64 ff. They also devote in the appendices about 20 pp to numerical
solutions / simulations. They only devote a few pp. to chaotic
oscillations. The newer versions of Marian (Thornton and Marian) do
much more on chaos. If interactive physics is a "black box" I suggest
substituting for your study, algorithms you write, thereby, possibly,
more easily checking the validity of your results.
John Denker wrote:
On 12/29/2007 12:26 PM, Ludwik Kowalski wrote:
The situation, however, seems to be paradoxical. On one hand we know
that undisturbed cycles are highly reproducible, on the other hand, we
see that a disturbance-due change, for example in the period T, is not
corrected after the disturbance is removed.
In physics, the definition of /chaos/ (i.e. deterministic chaos)
is extreme sensitivity to initial conditions.
So what is my point? I want to know how to use simulations, or any
other simple method, in a disagreement about stability or instability
I suspect stability and instability are the wrong concepts, or at
least the wrong terminology.
You might be much better off asking about chaotic versus non-chaotic.
There are simple yet powerful ways of studying chaotic systems, and
studying the transition from order to chaos. This was very trendy
in the 1980s.
http://www.pa.msu.edu/~bauer/applets/Chaos-Feigenbaum/feig.html
http://www.around.com/chaos.html
http://en.wikipedia.org/wiki/Julia_set
Once you understand what chaos is, the three-body orbit problem is
easily shown to be chaotic:
http://www.physics.drexel.edu/~steve/triple.html
=================================
Since the Subject: line asks about numerical simulations: it is
super-easy to get into a situation where a physics /simulation/
is numerically unstable, even if the underlying physics is well
behaved.
There are many bugs that can crop up in a numerical simulation,
which we can discuss if anybody is interested.
The Kepler problem is a remarkably good incubator for such bugs.
This is partly because we have such a good analytical (non-
numerical) solution to the problem. Numerical methods that are
not specifically tuned to this problem are almost guaranteed to
get the wrong answer. Energy won't be conserved, angular momentum
won't be conserved, and/or the orbital axis will precess when it
shouldn't.
These bugs can be dealt with to some extent, but the overall
problem is still a topic of research
http://www.springerlink.com/content/g3v21882l4042144/