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Helmholtz Differential Equation--Parabolic Coordinates

The scale factors are h_u=h_v=sqrt(u^2+v^2), h_theta=uv and the separation functions are f_1(u)=u, f_2(v)=v, f_3(theta)=1, given a Stäckel determinant of S=u^2+v^2. The Laplacian is

1/(u^2+v^2)(1/u(partialF)/(partialu)+(partial^2F)/(partialu^2)+1/v(partialF)/(partialv)+(partial^2F)/(partialv^2))+1/(u^2v^2)(partial^2F)/(partialtheta^2)+k^2F=0.
(1)

Attempt separation of variables by writing

F(u,v,theta)=U(u)V(v)Theta(theta),
(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.
(3)

Now multiply through by u^2v^2/(UVTheta),

(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.
(4)

Separating the Theta part gives

1/Theta(d^2Theta)/(dtheta^2)=-m^2,
(5)

which has solution

Theta(theta)=A_mcos(mtheta)+B_msin(mtheta).
(6)

Plugging (5) back into (4) and multiplying by (u^2+v^2)/(u^2v^2) 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).
(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).
(8)

This can be rearranged into two terms, each containing only u or v,

[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)]
(9)

and so can be separated by letting the first part equal c and the second equal -c, giving

(d^2U)/(du^2)+1/u(dU)/(du)+(k^2u^2-(m^2)/(u^2)-c)U=0
(10)
(d^2V)/(dv^2)+1/v(dV)/(dv)+(k^2v^2-(m^2)/(v^2)+c)V=0.
(11)

See also

Helmholtz Differential Equation, Parabolic Coordinates

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References

Arfken, G. "Parabolic Coordinates (xi,eta,phi)." §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111, 1970.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.

Cite this as:

Weisstein, Eric W. "Helmholtz Differential Equation--Parabolic Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationParabolicCoordinates.html

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