Last week, we looked at solving Laplace’s equation, in three dimensions, specifically Cartesian and cylindrical coordinates. Now, let us consider spherical coordinates (*ρ*,*φ*,*θ*) (we use the physicist’s convention of polar angle *θ* and azimuthal angle *φ*).

In spherical coordinates, Laplace’s equation is

Trying a solution of the form *f*(*ρ*,*φ*,*θ*)=*R*(*ρ*)*Φ*(*φ*)*Θ*(*θ*), we have:

The first bracket depends only on *ρ*, while the second depends only on the angular coordinates. We choose a constant to which the first bracket equals:

The angular portion is then:

and we have separated *φ* and *θ* terms. Expecting solutions that are continuous and periodic in 0≤*φ*<2π, we set

where m is an integer.

This gives us the remaining differential equation

.

To examine the solutions to this last equation, we first consider the *m*=0 case. Here, we have:

.

Let us make the change of variables . Then we have

This is amenable to solution by the method of Frobenius. One finds that the resulting power series converges for all -1<*x*<1 (0<*θ*<π), but that it diverges at *x*=±1 (blows up at the polar axis) unless the series terminates. As the recursive formula for the coefficients of the power series is , the polynomial terminates only if *k*=*l*(*l*+1), where *l* is a non-negative integer. Then the solution is a polynomial with degree *l*, which is an even polynomial for even *l* and an odd polynomial for odd *l*. The initial coefficient, *a*_{0} or *a*_{1}, is arbitrary, and is usually chosen so that . These polynomials are known as Legendre polynomials.

They can be generated via Rodrigues’ formula:

.

The first few polynomials are:

.

Now, for *m*≠0, we have, using *k*=*l*(*l*+1) (here *l* is not necessarily an integer),

,

called the “general Legendre equation.” It has solutions called the associated Legendre functions, written , where *l* is called the degree and *m* the order of the function. We already have that

, the legendre polynomial of degree *l*, when *l* is a non-negative integer.

It can be shown that the associated Legendre functions are well behaved on -1≤*x*≤1 for *l* and *m* integers with |*m*|≤*l*, *l*≥0, with the solutions for non-negative m *m* in terms of Legendre polynomials being

,

and the negative *m* solutions given by

.

(I have used here the Condon-Shortley phase.)

So our solutions will generally have a polar angle component

.

If we combine these with the azimuthal angular component written in terms of complex exponentials, we then have:

where *l* and *m* are integers with *l*≥0 and |*m*|≤*l*. Choosing our constant to produce the othonormal relation:

(where the integration is over all solid angles, ^{*} indicates the complex conjugate, and is the Kronecker Delta),

we get

.

These are called spherical harmonics.

The first few are:

Now, for our radial component, we have ; with *k*=*l*(*l*+1), we get

Trying a solution of the form , we find

or

So our radial component is of the form , and our solutions to Laplace’s equation in spherical coordinates will be of the form

.

Tags: Laplace's Equation, Math, Monday Math, Separation of Variables, Spherical Coordinates, Spherical Harmonics

January 5, 2009 at 12:05 am |

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January 21, 2009 at 6:46 am |

thanks for the info !

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