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 it obeys 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)

(Morse and Feshbach 1953, p. 509). This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666) give not only the Stäckel determinants for common coordinate systems, but also the elements of the determinant (although it is not clear how these are derived).

REFERENCES:

Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 5-7, 1988.

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.