“Cayley-Klein Parameters”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cayley-Klein Parameters

释义

Cayley-Klein Parameters

The parameters , , , and which, like the three Euler Angles, providea way to uniquely characterize the orientation of a solid body. These parameters satisfy the identities

			(1)
			(2)
			(3)
			(4)
			(5)

and

			(6)
			(7)

where

denotes the Complex Conjugate. In terms of the Euler Angles

, and

, the Cayley-Klein parameters are given by

			(8)
			(9)
			(10)
			(11)

(Goldstein 1960, p. 155).

The transformation matrix is given in terms of the Cayley-Klein parameters by

(12)

(Goldstein 1960, p. 153).

The Cayley-Klein parameters may be viewed as parameters of a matrix (denoted Q for its close relationship withQuaternions)

(13)

which characterizes the transformations

			(14)
			(15)

of a linear space having complex axes. This matrix satisfies

(16)

where I is the Identity Matrix and

the Matrix Transpose, as well as

(17)

In terms of the Euler Parameters

and the Pauli Matrices

, the Q-matrix can be written as

(18)

(Goldstein 1980, p. 156).

See also Euler Angles, Euler Parameters, Pauli Matrices, Quaternion

References

Goldstein, H. ``The Cayley-Klein Parameters and Related Quantities.'' §4-5 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 148-158, 1980.

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