Chapter

Hamiltonsche Quaternionen

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Abstract

Sir William Rowan Hamilton (geb. 1805 in Dublin; liest mit 5 Jahren Lateinisch, Griechisch und Hebräisch; 1823 Student am Trinity College in Dublin; 1827 als Undergraduate Student Berufung zum Professor der Astronomie an der Universität Dublin und zum Direktor der Sternwarte von Dunsink mit dem Titel: Royal Astronomer of Ireland; entwickelt 1827 die geometrische Optik aus Extremalprinzipien; 1834/35 Übertragung der Extremalprinzipien auf die Dynamik, Hamiltonsches Prinzip der kleinsten Wirkung, Hamiltonsche Wirkungsfunktion, kanonische Bewegungsgleichungen; 1835 Ritterschlag; 1837–1845 Präsident der Royal Irish Academy; 1843 Erfindung der Quaternionen; gest. 1865 in Dunsink) hat 1835 das Rechnen mit komplexen Zahlen x + iy logisch gerechtfertigt als ein Operieren mit geordneten reellen Zahlenpaaren (x, y) nach postulierten Rechenregeln (vgl. 3.1.8). Dies war der Ausgangspunkt für Hamiltons Interesse an der Frage, ob die geometrische Deutung des Addierens und vor allem des Multiplizierens von komplexen Zahlen in der Ebene ℝ2 nicht irgendwie — durch Schaffung hyperkomplexer Zahlen — auch im Anschauungsraum ℝ3 ein Analogon haben könne.

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... Now, H 4 has a natural quaternionic interpretation which arises as follows. The group of norm 1 units of the real quaternion algebra is easily identifiable with SU(2), see [22] for background material and notation. Using the 2-fold cover of SO(3) by ...
... Let us now briefly describe how the quaternions enter our (mainly geometric) picture, and how they provide a parametrization of (S)O(4) = (S)O(4, R), see [22] for details. The key is that pairs of quaternions in H(R), i.e. quaternions q = (q 0 , q 1 , q 2 , q 3 ) as written in the standard basis 1, i, j, k of the quaternion algebra over R, induce an action on vectors of R 4 via ...
... Consequently, when |q 1 | = |q 2 | = 1, we obtain a 4D rotation matrix and the homomorphism M : S 3 × S 3 −→ SO(4) provides the standard double cover of the rotation group SO(4) [22], with M(q 1 , q 2 ) = M(−q 1 , −q 2 ). The orientation reversing transformations, i.e. the elements of O(4)\SO(4), are obtained by the mapping x → q 1 x q 2 with unit quaternions q 1 , q 2 . ...
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