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John Creighto
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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
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
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