“Stäckel Determinant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Stäckel Determinant

释义

Stäckel Determinant

A Determinant used to determine in which coordinate systems the Helmholtz Differential Equation is separable(Morse and Feshbach 1953). A determinant

(1)

in which

are functions of

alone is called a Stäckel determinant. A coordinate system is separable if itobeys the Robertson Condition, namely that the Scale Factors

in the Laplacian

(2)

can be rewritten in terms of functions

defined by

(3)

such that

can be written

(4)

When this is true, the separated equations are of the form

(5)

The

s obey the minor equations

			(6)
			(7)
			(8)

which are equivalent to

(9)

(10)

(11)

This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666) give not only the Stäckeldeterminants for common coordinate systems, but also the elements of the determinant (although it is not clear how these arederived).

See also Helmholtz Differential Equation, Laplace's Equation, Poisson's Equation,Robertson Condition, Separation of Variables

References

Morse, P. M. and Feshbach, H. ``Tables of Separable Coordinates in Three Dimensions.'' Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 509-511 and 655-666, 1953.

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