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علماء الرياضيات | Helmholtz Differential Equation--Parabolic Coordinates |
 1485
 02:54 مساءً
date: 18-7-2018 |
Author : Moon, P. and Spencer, D. E 
Book or Source : Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag 
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The scale factors are 
, 
and the separation functions are 
, 
, 
, given a Stäckel determinant of 
. The Laplacian is
 |
(1)
|
Attempt separation of variables by writing
 |
(2)
|
then the Helmholtz differential equation becomes
![1/(u^2+v^2)[VTheta(1/u(dU)/(du)+(d^2U)/(du^2))+UTheta(1/v(dV)/(dv)+(d^2V)/(dv^2))]
+(UV)/(u^2v^2)(d^2Theta)/(dtheta^2)+k^2UVTheta=0.](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation3.gif) |
(3)
|
Now multiply through by 
,
![(u^2v^2)/(u^2+v^2)[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
+1/Theta(d^2Theta)/(dtheta^2)+k^2u^2v^2=0.](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation4.gif) |
(4)
|
Separating the 
part gives
 |
(5)
|
which has solution
 |
(6)
|
Plugging (5) back into (4) and multiplying by 
gives
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
-m^2(u^2+v^2)/(u^2v^2)+k^2(u^2+v^2).](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation7.gif) |
(7)
|
Rewriting,
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))]
-m^2(1/(v^2)+1/(u^2))+k^2(u^2+v^2).](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation8.gif) |
(8)
|
This can be rearranged into two terms, each containing only 
or 
,
![[1/U(1/u(dU)/(du)+(d^2U)/(du^2))+k^2u^2-(m^2)/(u^2)]
+[1/V(1/v(dV)/(dv)+(d^2V)/(dv^2))+k^2v^2-(m^2)/(v^2)]](http://mathworld.wolfram.com/images/equations/HelmholtzDifferentialEquationParabolicCoordinates/NumberedEquation9.gif) |
(9)
|
and so can be separated by letting the first part equal 
and the second equal 
, giving
 |
(10)
|
|
REFERENCES: Arfken, G. "Parabolic Coordinates Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York McGraw-Hill, pp. 514-515 and 660, 1953. |
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التوتر والسرطان.. علماء يحذرون من "صلة خطيرة"
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مرآة السيارة: مدى دقة عكسها للصورة الصحيحة
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نحو شراكة وطنية متكاملة.. الأمين العام للعتبة الحسينية يبحث مع وكيل وزارة الخارجية آفاق التعاون المؤسسي
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." §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111, 1970.
