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Posted by Ken S. Tucker on November 28, 2006, 2:59 am
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rob wrote:
> I'm quite enjoying this discussion on stability....a whole bunch of concepts
> here that I had not considered :)
> Thanks everyone!
Williams is a real sharp SOB, he's been helping me
with science since the late 70's, he's a genuine
scientist, he's 99% right except when he's wrong :-).
Ken
> >
> > David Williams wrote:
> >> -> But using the greatest supercomputer, you can't
> >> -> achieve mathematical perfection because you
> >> -> will always have a limited number of digits.
> >>
> >> True. But can you show that it's generally true that round-off errors
> >> will cause the simulated system to become unstable?
> >
> > Well of course, it's rather like boosting "dt".
> > Using differing dt's changes the result.
> >
> >> The Trojan asteroids are not exactly at the L-points on Jupiter's
> >> orbit. They are somewhat displaced in both location and velocity. But
> >> the situation is stable against minor perturbations. The asteroids
> >> oscillate around the L-points, never straying far from them.
> >
> > You have an oscillation radius for the Trojans
> > I'll denote Tr. Suppose Tr increases, while Jupiter
> > moves closer to the Sun. As Jupiter's orbital velocity
> > increases, the Tr will increase, eventually being
> > expelled from the L-point you mention, even possibly
> > toward Earth. That's a very interesting prospect.
> >
> >> Numerical "perturbations" might similarly be unimportant to the
> >> stability of the simulated system.
> >
> > There is a calculus of errors, and as you well know
> > I've done hundreds (1000's) of simulations, and there
> > is a means to partially cancel digitally induced errors,
> > however a residual error remains.
> > Recall 10 years ago, I ran Venus-asteroid sims for you
> > and the actual sim I used I pumped to higher dt's until
> > an evident failure resulted, what does that mean.
> > Well in my mind it means any finite dt has a residual
> > error, and an infinitesmal dt is impossible.
> > Even an analog computer will degrade to electron
> > noise, because at it's lowest level we're moving
> > electrons quantum mechanically, and pushing
> > the Heisenberg Uncertainty Principle (HUP) limit.
> >
> > So we're left with falling back onto pure theory
> > to prove a system is stable for n>2.
> > It's your turn.
> > Regards
> > Ken
> >
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