[FOM] The n-body problem

Alasdair Urquhart urquhart at cs.toronto.edu
Sun Apr 16 09:42:58 EDT 2006

Gabriel Stolzenberg wrote:

>    In a preprint I once received, a prominent philosopher wrote,
>     "As long as we accept the correctness of Newton's Law of Gravity,
>     for example, we are committed to the statement that the evolution
>     of an N-body system will be in accordance with the solutions to
>     the appropriate system of differential equations; and it is to
>     this day quite unknown whether the solutions to these equations
>     are recursively calculable even when N = 3."
>    The view he expresses here had been folklore since the beginning
> of the last century and, for all I know, still is.  However, I wrote
> back explaining that he was talking here about a 1st order ODE that,
> sufficiently close to the initial conditions (close enough to stay
> away from collisions and then some), satisfies a Lipshitz condition.
> Hence, we're talking about Picard's method, which is a construction
> par excellance.  (In this case, the solutions are even analytic.)
>    The philosopher then removed the statement from his paper.

I am puzzled by these remarks.  If I understand
what I've read in the semi-popular literature about
dynamical systems and celestial mechanics (for example,
"Celestial Encounters" by Diacu and Holmes), then no analytic
solution to the N-body problem is possible in general.
The existence of singularities in the solution space
(not just singularities caused by collisions, but also
noncollision singularities) seems to rule out any simple
answer to the problem.

Alasdair Urquhart

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