# Newtons Law of Gravity Legendre Polynomial & Harmonic functions

• John Creighto
In summary, section 17 of Mathematical Physics by Donald H. Menzel discusses the use of Legendre polynomials in physics, specifically for harmonic functions with a 1/r dependence. The author starts with Newton's law of gravitation and uses the law of cosines to express the distance from a point to the origin and another point in terms of the distance between the two points. Substitutions are then made to expand the equation using the binomial theorem. The coefficients in the expanded equation are represented as a series of Legendre polynomials, which can be derived from Laplace's equation in spherical coordinates. Menzel's method involves recognizing patterns in the expanded terms, while other books may use the last formula to define the 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:

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

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

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

Last edited:
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.

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.

## 1. What is Newton's Law of Gravity?

Newton's Law of Gravity is a fundamental law of physics that describes the force of gravity between two objects. It states that the force of gravity is directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance between them.

## 2. What are Legendre Polynomials?

Legendre Polynomials are a set of orthogonal polynomials that are often used in mathematics and physics to represent functions in terms of a series of polynomials. They are named after the French mathematician Adrien-Marie Legendre who first studied them.

## 3. How are Legendre Polynomials related to Newton's Law of Gravity?

Legendre Polynomials are used in physics to represent the force of gravity between two objects using a series of polynomials. They are used to solve for the gravitational potential and force between two objects, as described by Newton's Law of Gravity.

## 4. What are Harmonic Functions?

Harmonic Functions are mathematical functions that satisfy the Laplace's equation, which is a second-order partial differential equation. They are important in physics and engineering as they describe the behavior of many physical systems, such as vibrations and waves.

## 5. How are Harmonic Functions related to Newton's Law of Gravity?

Harmonic Functions are often used to model the gravitational potential and force between two objects, as described by Newton's Law of Gravity. This is because the gravitational force is a conservative force, and conservative forces can be described by harmonic functions.

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