Warum in der Theorie der dynamischen Systeme Kühe symmetrisch sein dürfen.

Wenn Mathematiker oder Physiker Differentialgleichungen auf einer Sphäre lösen wollen, nehmen sie gerne zunächst einmal an, dass diese rund und symmetrisch ist – das vereinfacht die Rechnungen oft enorm. Falls die Sphäre nun in Wirklichkeit nicht rund und symmetrisch ist, hat man jedenfalls eine Koordinatentransformation (einen Diffeomorphismus) zu einer runden Sphäre und auch wenn die exakten Lösungen natuerlich von der gewählten Transformation abhängen, bleiben manche topologische Eigenschaften der Lösungen bei der Transformation erhalten.


Gewöhnliche Differentialgleichungen kann man als Vektorfelder auffassen (TvF 202), die Nullstellen des Vektorfeldes entsprechen dann den stationären Punkten der Differentialgleichung.

In der Theorie der dynamischen Systeme interessiert man sich weniger für explizite Lösungen der Differentialgleichung als für das qualitative Verhalten der Orbits. Diese Herangehensweise geht vor allem auf Poincaré zurück und sie ermöglicht den Einsatz topologischer Methoden beim Studium dynamischer Systeme. Zum Beispiel kann man mit Hilfe des Jordanschen Kurvensatzes den Satz von Poincaré-Bendixsson beweisen, der die möglichen ω-Limesmengen für den Fluß einer Differentialgleichung auf der Ebene beschreibt.

In den letzten Wochen hatten wir über den gekämmten Igel und Vektorfelder auf der 2-dimensionalen Sphäre geschrieben und daß sich die Euler-Charakteristik einer Fläche aus den Singularitäten eines Vektorfelds berechnen läßt. Für die Sphäre z.B. war die Summe der Indizes der Nullstellen immer 2, man kann also zwei Nullstellen vom Index 1 haben (wie im 1. und 2.Bild) oder eine Nullstelle vom Index 2 (wie im 3.Bild).

Bemerkenswert am Satz von Hopf-Poincaré (daß die Summe der Indizes gleich der Euler-Charakteristik ist) ist natürlich, daß er für jedes Vektorfeld auf einer Fläche gilt und daß er auch überhaupt nicht davon abhängt, welche geometrische Form die Fläche konkret hat – es kommt nur auf die Topologie der Fläche an.
Statt irgenwie verbeulten Sphären braucht man also nur Vektorfelder auf einer runden Sphäre zu betrachten.



Die amerikanischen scienceblogs hatten im August mal einen Artikel “Spherical waves and the hairy ball” ueber die Neigung der Physiker, physikalische Sachverhalte am Modell einer (maximal symmetrischen) Sphäre zu untersuchen. (“Assume a spherical cow“, wir hatten das Thema damals in “Kühe in der Wissenschaft” aufgegriffen.)
Der Autor diskutierte das Beispiel sphärischer Wellen:

… what about spherically propagating waves? Light waves propagate perpendicular to the E and B fields, so perhaps we can have a light wave propagating radially outward. The E and B fields would be tangent to the outgoing spherical waves, skirting the problem that you can’t have non-static E and B fields both pointing radially outward.

Unfortunately we run into a brick wall there too. There’s a theorem in mathematics which states given a vector field defined on a sphere such that every vector is tangent to that sphere, the vector field must be zero on at least one point on that sphere. This theorem is called the hairy ball theorem. I’m not making this up. Basically if you have a basketball covered in fur, there’s no way to comb the entire thing smoothly. You’ll always have at least one cowlick.

But if either one of the E or B fields is zero at some point, the Poynting vector ExB will be zero too, meaning no light is being radiated from that point. Thus a spherical light wave is impossible too, not just in practice but in theory as well. Sometimes that isn’t a problem. We might be interested only in some region in which the assumption of spherical symmetry works just fine. Still, we are constrained …

Für diese spezielle Anwendung braucht man nur, daß es auf einer runden Sphäre immer Nullstellen zu einem Vektorfeld geben muß. Aber, wie gesagt, dasselbe wäre damit dann auch der Fall auf völlig unsymmetrischen Sphären:

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