Stäckel Determinant
المؤلف:
Moon, P. and Spencer, D. E
المصدر:
Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
...
25-7-2018
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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
![1/(h_1h_2h_3)partial/(partialu_i)((h_1h_2h_3)/(h_i^2)partial/(partialu_i))
=(g(u_(i+1),u_(i+2)))/(h_1h_2h_3)partial/(partialu_i)[f_i(u_i)partial/(partialu_i)]
=1/(h_i^2f_i)partial/(partialu_i)(f_ipartial/(partialu_i))](http://mathworld.wolfram.com/images/equations/StaeckelDeterminant/NumberedEquation3.gif) |
(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
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.
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