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Laplace Equation--Confocal Ellipsoidal Coordinates

Using the notation of Byerly (1959, pp. 252-253), Laplace's equation can be reduced to

del ^2F=(mu^2-nu^2)(partial^2F)/(partialalpha^2)+(lambda^2-nu^2)(partial^2F)/(partialbeta^2)+(lambda^2-mu^2)(partial^2F)/(partialgamma^2)=0,
(1)

where

alpha=cint_c^lambda(dlambda)/(sqrt((lambda^2-b^2)(lambda^2-c^2)))
(2)
=F(b/c,pi/2)-F(b/c,sin^(-1)(c/lambda))
(3)
beta=cint_b^mu(dmu)/(sqrt((c^2-mu^2)(mu^2-b^2)))
(4)
=F(sqrt(1-b^2-c^2),sin^(-1)(sqrt((1-(b^2)/(mu^2))/(1-(b^2)/(c^2)))))
(5)
gamma=cint_0^nu(dnu)/(sqrt((b^2-nu^2)(c^2-nu^2)))
(6)
=F(b/c,sin^(-1)(nu/b)).
(7)

In terms of alpha, beta, and gamma,

lambda=cdc(alpha,b/c)
(8)
mu=bnd(beta,sqrt(1-(b^2)/(c^2)))
(9)
nu=bsn(gamma,b/c).
(10)

Equation (◇) is not separable using a function of the form

F=L(alpha)M(beta)N(gamma),
(11)

but it is if we let

1/L(d^2L)/(dalpha^2)=suma_klambda^k
(12)
1/M(d^2M)/(dbeta^2)=sumb_kmu^k
(13)
1/N(d^2N)/(dgamma^2)=sumc_knu^k.
(14)

These give

a_0=-b_0=c_0
(15)
a_2=-b_2=c_2,
(16)

and all others terms vanish. Therefore (◇) can be broken up into the equations

(d^2L)/(dalpha^2)=(a_0+a_2lambda^2)L
(17)
(d^2M)/(dbeta^2)=-(a_0+a_2mu^2)M
(18)
(d^2N)/(dgamma^2)=(a_0+a_2nu^2)N.
(19)

For future convenience, now write

a_0=-(b^2+c^2)p
(20)
a_2=m(m+1),
(21)

then

(d^2L)/(dalpha^2)-[m(m+1)lambda^2-(b^2+c^2)p]L=0
(22)
(d^2M)/(dbeta^2)+[m(m+1)mu^2-(b^2+c^2)p]M=0
(23)
(d^2N)/(dgamma^2)-[m(m+1)nu^2-(b^2+c^2)p]N=0.
(24)

Now replace alpha, beta, and gamma to obtain

(lambda^2-b^2)(lambda^2-c^2)(d^2L)/(dlambda^2)+lambda(lambda^2-b^2+lambda^2-c^2)(dL)/(dlambda) -[m(m+1)lambda^2-(b^2+c^2)p]L=0 (mu^2-b^2)(mu^2-c^2)(d^2M)/(dmu^2)+mu(mu^2-b^2+mu^2-c^2)(dM)/(dmu) -[m(m+1)mu^2-(b^2+c^2)p]M=0 (nu^2-b^2)(nu^2-c^2)(d^2N)/(dnu^2)+nu(nu^2-b^2+nu^2-c^2)(dN)/(dnu) -[m(m+1)nu^2-(b^2+c^2)p]N=0.
(25)

Each of these is a Lamé's differential equation, whose solution is called an ellipsoidal harmonic of the first kind. Writing

L(lambda)=E_m^p(lambda)
(26)
M(lambda)=E_m^p(mu)
(27)
N(lambda)=E_m^p(nu)
(28)

gives the solution to (◇) as a product of ellipsoidal harmonics of the first kind E_m^p(x).

F=E_m^p(lambda)E_m^p(mu)E_m^p(nu).
(29)

See also

Confocal Ellipsoidal Coordinates, Ellipsoidal Harmonic of the First Kind, Helmholtz Differential Equation

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References

Arfken, G. "Confocal Ellipsoidal Coordinates (xi_1,xi_2,xi_3)." §2.15 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 117-118, 1970.Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251-258, 1959.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 43-44, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.

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Laplace Equation--Confocal Ellipsoidal Coordinates

Cite this as:

Weisstein, Eric W. "Laplace Equation--Confocal Ellipsoidal Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LaplaceEquationConfocalEllipsoidalCoordinates.html

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