Re: Surviving Einstein.

From: Paul Schlyter (pausch_at_saaf.se)
Date: 07/31/03 

Date: Wed, 30 Jul 2003 22:43:16 +0000 (UTC)

In article <3F279476.6090305@hidden.com>, AE <nospam@hidden.com> wrote:
 
> Paul Schlyter wrote:
>> AE wrote:
>>>Paul Schlyter wrote:
>>>>An analytic solution covers a very wide range of cases in one and the
>>>>same solution, while a numeric solution only covers that specific case.
>>>>...
>>>>Suppose you want to find the area of a circle. What method would
>>>>you prefer?
>>>
>>>Surely I know analytic solutions are to be prefered due to their
>>>generality.
>>>
>>>Nevertheless a numeric solution is a solution: In the end I'll get the same
>>>area for my circle, independent of the way I'm using to calculate it.
>>
>> No you don't....
>>
>> If the radius of the circle is 1 exactly, the analytical solution will
>> quickly tell you that the area is pi exactly.
>>
>> The numerical solution will be unable to tell you the exact area. All
>> you can get from the numerical solution is an approximate area: 3.14159....
>
> That's absolutely correct for a theoretic object.
>
> When handling real-world problems we are anyway working with
> approximations - or does anyone want to calculate the precise shape of
> an electron orbit due to the geometric changes caused by the gravity of
> earth? And if yes: For what purpose?
>
>> Yes, there's a difference between these two solutions.
>
> I do agree with you that there's a difference in theory. For all
> practical purposes (and here it doesn't matter whether it is the
> description of an existing or the prediction of future results) this
> doesn't matter at all.
 
If we consider the problem of planetary motions in the solar system,
then the presence of an analytical solution will be able to tell us
whether there are any inherent instability in the orbits or not. A
numerical solution cannot tell us that, unless an instability is
encountered within the time span of the integration of course. But
the absence of such instabilities during the time span of the
integration says little about the absence of instabilities during
much longer time spans. And our best numerical integrations cover
time spans which are only a fraction of the age of the solar system,
as well as of its expected remaining lifetime.
 
But perhaps you'll argue that the long term stability of the solar
system is completely uninteresting to us, because it'll remain
stable within our lifetimes anyway????
  

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