Expanding Newton's gravity potential using Legendre polynomials

[John Creighto]

I'm reading section 17 of Mathematical Physics by Donald H. Menzel on Harmonic functions.

They start with Newtons law of gravitation (although the following method can be aplied to any potential field with a 1/r dependence.)
see: http://en.wikipedia.org/wiki/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics

$$dV=-\frac{GdM}{r_{12}}=\frac{GdM}{(r_2^2-2r_1r_2 cos \gamma + r_1^2)^{1/2}}$$

Where:
$$R_{12}$$=The distance from P1 to the unit volume
$$R_{1}$$ is the distrance from the orgin to the unit volume
$$R{2}$$ is the distance from P1 to the orgin.
note that the law of cosines was used to express $$R_{12}$$ in terms of $$R_1$$ and $$R_2$$

The substitutions:

$$cos \gamma = \mu$$
$$\frac{r_1}{r_2}=\Beta_1$$

are made giving:

$$dV=-\frac{GdM}{r_2} \left[ 1 - \beta (2 \mu - \beta) \right]^{-1/2}$$

This is expanded using the more general form of the binomial theorem:$$dV=-\frac{GdM}{r_2} \left[ 1 + \frac{1}{2}\beta (2\mu - \beta ) + \frac{1*2}{2*4}\beta^2(2\mu-\beta)^2+... \right]$$

Now here is I where I get lost. If you expand and collect the terms (powers of $$\beta$$) then supposedly the coefficients are:

$$P_l(\mu )=\frac{(2l)!}{2^l(l!)^2} \left[ u^l - \frac{l(l-1)}{2(2(l-1)}\mu^{l-2}+\frac{l(l-1)(l-2)(l-3)}{2 * 4(2l-1)(2l-3)}\mu^{l-4}... \right]$$

and apparently the numerical coefficients can be represented as follows:

$$\frac{(2l)!}{2^l(l!)^2}=\frac{(2l-1)(2l-3)...1}{l!}$$

This then gives:
$$dV=-\frac{GdM}{r_2} \sum_{l=0}^{\inf}P_l(\mu ) \left( \frac{r_1}{r_2} \right)^l$$

 

[pam]

Newer physics books give clearer derivations.
Laplace's equation in spherical coordinates, leads to Legendre's equation.
Then your last formula is a special case.
In Menzel's method, terms like (2mu-beta)^n have to be expanded in the binomial expansion,and then the pattern recognized.
Some books (Arfken) go backwards, using your last equation to defined the LPs.