The dramatic episode of Sundman

Abstract

In 1912 the Finnish mathematical astronomer Karl Sundman published a remarkable solution to the three-body problem, of a type which mathematicians such as Poincaré had believed impossible to achieve. Although lauded at the time, the result dimmed from view as the twentieth century progressed and its significance was often overlooked. This article traces Sundman's career and the path to his achievement, bringing to light the involvement of Ernst Lindelöf and Gösta Mittag-Leffler in Sundman's research and professional development, and including an examination of the reception over time of Sundman's result. A broader perspective on Sundman's research is provided by short discussions of two of Sundman's later papers: his contribution to Klein's Encyklopädie and his design for a calculating machine for astronomy. ZusammenfassungIm Jahre 1912 veröffentlichte der finnische mathematische Astronom, Karl Sundman eine bemerkenswerte Lösung des Dreikörperproblems, von einer Art, wie sie Mathematiker wie Poincaré für unmöglich gehalten hatten. Obwohl Sundmans Ergebnis in seiner Zeit gefeiert wurde, verschwand es im Laufe des Jahrhunderts allmählich aus dem wissenschaftlichen Bewusstsein, und seine Bedeutung wurde meistens unterschätzt. Der Artikel spürt der Karriere Sundmans nach bis hin zu seiner großen Leistung. Er zeigt den Einfluss von Ernst Lindelöf and Gösta Mittag-Leffler auf Sundmans Forschung und berufliche Entwicklung. Er beschreibt auch die wechselnde Rezeption des Ergebnisses. Zwei spätere Publikationen Sundmans werden in einem Anhang beschrieben.

Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

“…Note that (23) follows from (28), so ( 28) is a necessary and sufficient condition the metric (11) (with g = 1) to be flat. In addition, as Ω is x-independent, it follows from (10) that…”

“…For further analysis, it is useful to introduce a function Φ(t) such that Ω = dΦ dt and rewrite the second equation in (28) in the form…”

Searching for distinct classes of atomic and molecular states using convergence and separability criteria

“…The TBS is a very complex and challenging problem, and to describe the TBS here is outside the scope of this work. For our discussion, it is enough to consider the Sundman's theorem, which, in principle, can indicate the general solution of the three-body problem in terms of a summable infinite series with exceptionally slowly convergence [18,19]. This behavior of the series would imply a practical impossibility to obtain a general solution [20] because of involving N ~ 10 10000000 = (10↑) 2 (in Knuth's up-arrow notation) series terms [21].…”

Stability of trajectories from Poincaré to Birkhoff: approaching a qualitative definition

“…The historical context of these works and their reception have been analyzed in detail by Barrow-Green (2010). She observes that in the years following the 1912 publication, "while Sundman's mathematical dexterity was widely lauded, it had not gone unnoticed that his result gave no practical help to astronomers, whose agenda was rather different from that of the mathematicians" (Barrow-Green 2010, p. 191).…”