Poincare's last geometric theorem

Question

Which problem in celestial mechanics led Poincare to his conjecture about number of fixed points of area preserving maps of the annulus?

Answer 1

This is also known as the Poincare-Birkhoff (or Poincare's last geometric theorem). In his work on dynamics, Poincare-Birkhoff was led to focus attention primarily upon the periodic motions.

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He employed the method of analytic continuation in his great Prize Memoir in the Acta Mathematics which dealt with the problem of $n $-bodies. In the integrable limiting case when the masses of all but one of the bodies vanish, there are infinitely many periodic motions. By varying certain parameters, he passed from this trivial limiting case to the case that none of the masses are zero, and showed that these periodic motions persist as members of analytic families, unless two of them combine and disappear from the real domain like the roots of algebraic equations with real coefficients. He did not consider the possibility of disappearance of such a motion, by its period becoming infinite, although this possibility requires consideration also.

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Unfortunately this method of analytic continuation gave very meagre results due to: Although there are infinitely many periodic families, it is conceivable that the range of the parameters become less and less as the type of periodic motion becomes more and more complicated. This would mean that only a finite number of the periodic motions might exist for any particular set of values of the parameters other than that of the trivial integrable case.

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Thus Poincare found the method of analytic continuation to be insufficient and was forced to find other methods of attack. Notwithstanding this severe limitation, and despite many years of effort, Poincare was not able to fully attain his goal. This gave rise to the Poincare-Birkhoff theorem during his last years.