Newtons Law of Gravity Legendre Polynomial & Harmonic functions

[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.

 

[charng]

I am familiar with Newton's Law of Gravity and its application in various fields of physics. In this case, we are discussing its application in the study of harmonic functions, which are functions that satisfy Laplace's equation. This equation is important in mathematical physics because it arises in many different physical problems, including electrostatics, fluid mechanics, and heat conduction.

The use of Legendre polynomials in the expansion of the gravitational potential is a powerful tool in solving problems related to potential fields. These polynomials are a set of orthogonal functions that have been extensively studied and have many applications in physics, including in the study of spherical harmonics. In this case, we are using them to expand the potential in terms of the distance from the origin and the distance between two points.

By making the appropriate substitutions and expanding the potential using the binomial theorem, we can see that the coefficients are related to the Legendre polynomials. This is a powerful result, as it allows us to express the potential in terms of these well-studied functions, making it easier to solve problems related to potential fields.

Furthermore, the numerical coefficients can be represented in a simplified form, which is useful in calculations and allows us to easily see the relationship between the Legendre polynomials and the expansion of the potential. This result is important in the study of harmonic functions and can be applied to other potential fields as well.

In conclusion, the use of Legendre polynomials in the expansion of the gravitational potential is a valuable tool in the study of harmonic functions and potential fields. It allows us to express the potential in terms of well-studied functions and simplifies calculations, making it easier to solve problems related to potential fields. This is just one example of the many applications of Legendre polynomials in physics and mathematics.