Karl Sundman's solution to the 3-body problem
himog
> In <819d1153.01100...@posting.google.com> hi...@email.com
> (himog) writes:
> >Where can I get Karl Sundham's 1913 series solution to the
> >Newtonian 3-body problem?
> It's Sundman, not Sundham. I would look in some of the classical celestial
> mechanics books published in the early 20th century.
>
> As you may know, this solution is not useful for calculating
> orbits because the series converge very slowly. That's perhaps
> why it's not widely published.
I looked in about 20 older books on celestial mechanics and whatnot
in my library, and I could not find it. And yes, I already knew that
the series converges extremely slowly. I still need to see a complete
presentation of this solution in English for some research I'm doing,
so if anyone can help me find it I would be very thankful.
Paul D. Shocklee
> I looked in about 20 older books on celestial mechanics and whatnot
> in my library, and I could not find it. And yes, I already knew that
> the series converges extremely slowly. I still need to see a complete
> presentation of this solution in English for some research I'm doing,
> so if anyone can help me find it I would be very thankful.
I believe the following references should be helpful:
Florin Diacu, "The Solution of the n-body Problem," The Mathematical
Intelligencer, vol. 18, no. 3, 1996
"A visit to the Newtonian N-body problem via elementary complex variables,"
The American Mathematical Monthly 97 (1990), 105-119
Quidong Wang, "The global solution of the n-body problem," Celestial
Mechanics 50 (1991), 73-88
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Phone: (703) 704-9109
Chris Hillman
On 2 Oct 2001, himog (hi...@email.com) asked:
> Where can I get Karl Sundham's 1913 series solution to the Newtonian
> 3-body problem?
On 5 Oct 2001, John E. Prussing (prus...@uiuc.edu) replied:
> It's Sundman, not Sundham. I would look in some of the classical
> celestial mechanics books published in the early 20th century.
>
> As you may know, this solution is not useful for calculating orbits
> because the series converge very slowly. That's perhaps why it's not
> widely published.
???
If I am not mistaken, the paper in question is:
K. E. Sundman,
"Memoire sur le probleme de trois corps",
Acta Mathematica 36 (1912): 105--179.
I've never heard of "Sundman's expansion", although I've heard of many
other "named perturbative expansions" in the context of the three-body
problem (or N-body problem) in Newtonian mechanics, e.g. one sometimes
hears phrases like "Linstedt series", "secular expansion" (although
generally these arise ultimately from standard "orthonormal basis for a
seperable Hilbert space" expansions in terms of eccentricities or other
orbital parameters).
OTH, in the paper cited above, Sundman proves that binary collision can
occur only if the total angular momentum vanishes. This is called
"Sundman's theorem" and it is discussed in most monographs on advanced
celestial mechanics. In particular, the theorem and its implications are
extensively discussed in the huge book
Yusuke Hagihara,
Celestial Mechanics. (Vol I and Vol II pt 1 and Vol II pt 2.)
MIT Press, 1970.
A more recent book which discusses Sundman's theorem is
D. Boccaletti and G. Pucacco,
Theory of Orbits (two volumes).
Springer-Verlag, 1998.
HTH,
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
Chris Hillman
On Tue, 6 Nov 2001, I wrote:
> On 5 Oct 2001, John E. Prussing (prus...@uiuc.edu) replied:
>
> > As you may know, this solution is not useful for calculating orbits
> > because the series converge very slowly. That's perhaps why it's not
> > widely published.
>
> ???
>
> If I am not mistaken, the paper in question is:
>
> K. E. Sundman,
> "Memoire sur le probleme de trois corps",
> Acta Mathematica 36 (1912): 105--179.
>
> I've never heard of "Sundman's expansion", although I've heard of many
> other "named perturbative expansions"
Oh dear, by the time this retraction appears I probably will have been
quite properly raked over the coals, but anyway-- apologies to John
Prussing and to "himog" and thanks to my colleague Marshall Hampton (who
is finishing a dissertation on the N-body problem) for telling me that
Sundman did indeed provide a series expansion solution for the N-body
problem, which, however, suffers from severe convergence problems as John
Prussing noted.
Sorry for the confusion I inadvertently created.