Laplace's Equation--Spherical Coordinates

In spherical coordinates, the scale factors are , , , and the separation functions are , , , giving a Stäckel determinant of .

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

(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 ,

(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

(6)

which has solutions which may be defined either as a complex function with , ...,

(7)

or as a sum of real sine and cosine functions with , ...,

(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

(10)

(11)

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

(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)

(15)

This must hold true for all powers of . For the term (with ),

(16)

which is true only if and all other terms vanish. So for , . Therefore, the solution of the component is given by

(17)

Plugging (17) back into (◇),

(18)

(19)

which is the associated Legendre differential equation for and , ..., . 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

(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 can be absorbed by and , 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

(24)

(25)

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

(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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