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Special coordinates in the rotation group of three-dimensional space, the construction of which is based in the final analysis on the relationship between
and the group
of
unitary matrices with determinant 1. There exists a mapping
which is an epimorphism by virtue of its algebraic properties and a double covering by virtue of its topological properties. (Restricted to some neighbourhood of the identity matrix,
is an isomorphism; in other words,
and
are locally isomorphic.) Each matrix
may be written as
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where are complex numbers such that
. These are taken to be the Cayley–Klein parameters of
. (The term is sometimes used for all four elements of the matrix
.) As the actual construction of a mapping
with the above properties may be accomplished in various ways, different authors define the Cayley–Klein parameters in slightly different ways (see [2], [3]).
Since is not a true isomorphism, but only a double covering, it is impossible to define the Cayley–Klein parameters as (continuous) coordinates on all of
; this can be done only locally. However, the Cayley–Klein parameters may nevertheless be used to study processes of rotation in which
is a continuous function of a real parameter
(and there is no necessity to restrict the domain of possible values of
in any way). Indeed, if some fixed value
has been chosen at
, the corresponding values of
are uniquely defined by continuity for all
. (The fact that the complete inverse
is double-valued intrudes only in the observation that
not only when
but also when
.) Thus the Cayley–Klein parameters can be applied to investigate the motion of a rigid body with a fixed point (the configuration space of which is
). This approach was adopted in [1], but did not achieve popularity.
The group is isomorphic to the group of quaternions with norm 1 (cf. Quaternion); hence, by going over from
to the corresponding quaternion
, one can replace the Cayley–Klein parameters by the Euler–Rodriguez parameters — four real numbers
such that
. These stand in a simple relationship to the Cayley–Klein parameters (see [1], [3]) and possess the same "double-valuedness" property (for the history of the problem see [1]). It was essentially in this context that attention was first directed to two-valued representations of the rotation group (see Spinor).
References
[1] | F. Klein, A. Sommerfeld, "Ueber die Theorie des Kreises" , 1–2 , Teubner (1965) |
[2] | H. Goldstein, "Classical mechanics" , Addison-Wesley (1953) |
[3] | J.L. Synge, "Classical dynamics" , Handbuch der Physik , 3/1 , Springer (1960) pp. 1–225 |
Comments
References
[a1] | P. Du Val, "Homographies, quaternions and rotations" , Clarendon Press (1964) |
Cayley-Klein parameters. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Klein_parameters&oldid=22273