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Laplace's Equation--Spherical Coordinates

In spherical coordinates, the scale factors are h_r=1, h_theta=rsinphi, h_phi=r, and the separation functions are f_1(r)=r^2, f_2(theta)=1, f_3(phi)=sinphi, giving a Stäckel determinant of S=1.

The Laplacian is

del ^2=1/(r^2)partial/(partialr)(r^2partial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)partial/(partialphi)(sinphipartial/(partialphi)).
(1)

To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing

F(r,theta,phi)=R(r)Theta(theta)Phi(phi).
(2)

Then the Helmholtz differential equation becomes

(d^2R)/(dr^2)PhiTheta+2/r(dR)/(dr)PhiTheta+1/(r^2sin^2phi)(d^2Theta)/(dtheta^2)PhiR+(cosphi)/(r^2sinphi)(dPhi)/(dphi)ThetaR+1/(r^2)(d^2Phi)/(dphi^2)ThetaR=0.
(3)

Now divide by RThetaPhi,

(r^2sin^2phi)/(PhiRTheta)PhiTheta(d^2R)/(dr^2)+2/r(r^2sin^2phi)/(PhiRTheta)PhiTheta(dR)/(dr)+1/(r^2sin^2phi)(r^2sin^2phi)/(PhiRTheta)PhiR(d^2Theta)/(dtheta^2)+(cosphi)/(r^2sinphi)(r^2sin^2phi)/(PhiThetaR)(dPhi)/(dphi)ThetaR+1/(r^2)(r^2sin^2phi)/(PhiRTheta)(d^2Phi)/(dphi^2)ThetaR=0
(4)
((r^2sin^2phi)/R(d^2R)/(dr^2)+(2rsin^2phi)/R(dR)/(dr))+(1/Theta(d^2Theta)/(dtheta^2)) +((cosphisinphi)/Phi(dPhi)/(dphi)+(sin^2phi)/Phi(d^2Phi)/(dphi^2))=0.
(5)

The solution to the second part of (5) must be sinusoidal, so the differential equation is

(d^2Theta)/(dtheta^2)1/Theta=-m^2,
(6)

which has solutions which may be defined either as a complex function with m=-infty, ..., infty

Theta(theta)=A_me^(imtheta),
(7)

or as a sum of real sine and cosine functions with m=-infty, ..., infty

Theta(theta)=S_msin(mtheta)+C_mcos(mtheta).
(8)

Plugging (6) back into (7),

(r^2)/R(d^2R)/(dr^2)+(2r)/R(dR)/(dr)+1/(sin^2phi)((cosphisinphi)/Phi(dPhi)/(dphi)+(sin^2phi)/Phi(d^2Phi)/(dphi^2)-m^2)=0.
(9)

The radial part must be equal to a constant

(r^2)/R(d^2R)/(dr^2)+(2r)/R(dR)/(dr)=l(l+1)
(10)
r^2(d^2R)/(dr^2)+2r(dR)/(dr)=l(l+1)R.
(11)

But this is the Euler differential equation, so we try a series solution of the form

R=sum_(n=0)^inftya_nr^(n+c).
(12)

Then

r^2sum_(n=0)^infty(n+c)(n+c-1)a_nr^(n+c-2)+2rsum_(n=0)^infty(n+c)a_nr^(n+c-1) -l(l+1)sum_(n=0)^inftya_nr^(n+c)=0
(13)
sum_(n=0)^infty(n+c)(n+c-1)a_nr^(n+c)+2sum_(n=0)^infty(n+c)a_nr^(n+c) -l(l+1)sum_(n=0)^inftya_nr^(n+c)=0
(14)
sum_(n=0)^infty[(n+c)(n+c+1)-l(l+1)]a_nr^(n+c)=0.
(15)

This must hold true for all powers of r. For the r^c term (with n=0),

c(c+1)=l(l+1),
(16)

which is true only if c=l,-l-1 and all other terms vanish. So a_n=0 for n!=l, -l-1. Therefore, the solution of the R component is given by

R_l(r)=A_lr^l+B_lr^(-l-1).
(17)

Plugging (17) back into (◇),

l(l+1)-(m^2)/(sin^2phi)+(cosphi)/(sinphi)1/Phi(dPhi)/(dphi)+1/Phi(d^2Phi)/(dphi^2)=0
(18)
Phi^('')+(cosphi)/(sinphi)Phi^'+[l(l+1)-(m^2)/(sin^2phi)]Phi=0,
(19)

which is the associated Legendre differential equation for x=cosphi and m=0, ..., l. The general complex solution is therefore

sum_(l=0)^inftysum_(m=-l)^l(A_lr^l+B_lr^(-l-1))P_l^m(cosphi)e^(-imtheta) =sum_(l=0)^inftysum_(m=-1)^l(A_lr^l+B_lr^(-l-1))Y_l^m(theta,phi),
(20)

where

Y_l^m(theta,phi)=P_l^m(cosphi)e^(-imtheta)
(21)

are the (complex) spherical harmonics. The general real solution is

sum_(l=0)^inftysum_(m=0)^l(A_lr^l+B_lr^(-l-1))P_l^m(cosphi)[S_msin(mtheta)+C_mcos(mtheta)].
(22)

Some of the normalization constants of P_l^m can be absorbed by S_m and C_m, so this equation may appear in the form

sum_(l=0)^inftysum_(m=0)^l(A_lr^l+B_lr^(-l-1))P_l^m(cosphi)[S_l^msin(mtheta)+C_l^mcos(mtheta)] =sum_(l=0)^inftysum_(m=0)^l(A_lr^l+B_lr^(-l-1))×[S_l^mY_l^(m(o))(theta,phi)+C_l^mY_l^(m(e))(theta,phi)],
(23)

where

Y_l^(m(o))(theta,phi)=P_l^m(cosphi)sin(mtheta)
(24)
Y_l^(m(e))(theta,phi)=P_l^m(cosphi)cos(mtheta)
(25)

are the even and odd (real) spherical harmonics. If azimuthal symmetry is present, then Theta(theta) is constant and the solution of the Phi component is a Legendre polynomial P_l(cosphi). The general solution is then

F(r,phi)=sum_(l=0)^infty(A_lr^l+B_lr^(-l-1))P_l(cosphi).
(26)

See also

Helmholtz Differential Equation--Spherical Coordinates

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References

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, p. 244, 1959.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 27, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953.

Cite this as:

Weisstein, Eric W. "Laplace's Equation--Spherical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html

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