ODE: Legendre Polynomials

Legendre polynomials are not just abstract mathematical curiosities; they are the fundamental building blocks for solving a vast array of physical problems with spherical symmetry. From calculating the gravitational field of a planet to determining the electric potential around a charged sphere, these special functions provide the key to separating variables in the Laplace equation ${\nabla }^{2}V=0$ in spherical coordinates. Mastering their properties is essential for any engineer or physicist working in fields like electromagnetics, quantum mechanics, and fluid dynamics.

Derivation from the Legendre Equation

The journey to Legendre polynomials begins with attempting to solve Laplace's equation in spherical coordinates $(r,\theta ,\varphi )$. When a problem exhibits azimuthal symmetry (no dependence on the angle $\varphi$), the angular part of the separated differential equation reduces to the Legendre differential equation: $$(1-x^2)\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+l(l+1)y=0$$ Here, $x=\cos \theta$ (so $x\in [-1,1]$), and $l$ is a separation constant, often a non-negative integer. This is a second-order linear ODE with regular singular points at $x=\pm 1$. The method of power series (Frobenius method) reveals that for non-integer $l$, the solutions diverge at $x=\pm 1$, which is physically unacceptable for bounded potentials. However, when $l$ is an integer $0,1,2,...$, one of the two linearly independent series solutions terminates, becoming a polynomial. These finite, well-behaved polynomial solutions are the Legendre polynomials, denoted $P_l(x)$.

The first few Legendre polynomials, which form an orthogonal set, are: $P_0(x)=1$ $P_1(x)=x$ $P_2(x)=\frac{1}{2}(3x^2-1)$ $P_3(x)=\frac{1}{2}(5x^3-3x)$

Rodrigues Formula and Key Properties

While the series method defines these polynomials, a remarkably compact and powerful tool for generating them is the Rodrigues formula: $$P_l(x)=\frac{1}{2^{l}l!}\frac{d^l}{d{x}^l}(x^2-1{)}^l$$ This formula is incredibly useful for deriving properties and computing higher-order polynomials directly. For example, to find $P_2(x)$, you would compute the second derivative of $(x^2-1{)}^2=x^4-2x^2+1$, which yields $12x^2-4$, and then apply the factor $\frac{1}{(2^{2}2!)}=\frac{1}{8}$ to get $\frac{1}{2}(3x^2-1)$.

From Rodrigues formula, several critical properties follow:

• Parity: $P_l(-x)=(-1{)}^l{P}_l(x)$. Even-indexed polynomials are even functions, odd-indexed ones are odd.
• Normalization: $P_l(1)=1$ and $P_l(-1)=(-1{)}^l$.
• Recurrence Relations: These allow efficient computation. A common one is:

$$(l+1)P_{l+1}(x)=(2l+1)x{P}_l(x)-l{P}_{l-1}(x)$$ Knowing $P_0$ and $P_1$, you can generate all subsequent polynomials using this relation.

Orthogonality and Series Expansions

Perhaps their most important property for applications is orthogonality. On the interval $[-1,1]$, Legendre polynomials satisfy: $${\int }_{-1}^1{P}_m(x)P_n(x)dx=\frac{2}{2n+1}{\delta }_{mn}$$ where ${\delta }_{mn}$ is the Kronecker delta (1 if $m=n$, 0 otherwise). This means the integral of the product of two different Legendre polynomials is zero.

This orthogonality is the foundation for the Legendre series expansion. Just as any reasonable function can be expanded in a Fourier series of sines and cosines, any piecewise smooth function $f(x)$ defined on $[-1,1]$ can be expanded as a series of Legendre polynomials: $$f(x)=\sum _{l=0}^{\infty }{c}_l{P}_l(x)$$ The coefficients $c_l$ are found by exploiting orthogonality, analogous to Fourier coefficients: $$c_l=\frac{2l+1}{2}{\int }_{-1}^{1}f(x)P_l(x)dx$$ This technique, a specific type of Sturm-Liouville theory, is called eigenfunction expansion. It is the primary method for solving boundary value problems where the boundary condition is given on a spherical surface.

The Generating Function

A different but equally valuable perspective is provided by the generating function for Legendre polynomials: $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum _{l=0}^{\infty }{P}_l(x)t^l,\text{for }|t|<1$$ The function on the left "generates" the polynomials as the coefficients in its power series expansion in $t$. This formulation is not just a neat trick; it has direct physical significance. If you set $t=r_{<}/r_{>}$ and $x=\cos \gamma$, where $r_{<}$ and $r_{>}$ are the smaller and larger distances from the origin to two points and $\gamma$ is the angle between them, the generating function becomes the key to multipole expansions.

Applications to Gravitational and Electrostatic Potentials

The generating function leads directly to the flagship application: expanding the inverse distance term in potential theory. In both Newtonian gravity and electrostatics, the potential involves a term $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$. $$\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}=\frac{1}{r_{>}}\frac{1}{\sqrt{1-2(r_{<}/r_{>})\cos \gamma +(r_{<}/r_{>}{)}^2}}=\frac{1}{r_{>}}\sum _{l=0}^{\infty }{\left(\frac{r_{<}}{r_{>}}\right)}^l{P}_l(\cos \gamma )$$ This multipole expansion is profound. It allows the complex potential from an arbitrary charge or mass distribution to be broken down into a sum of simpler contributions: the $l=0$ term is the monopole (like a point charge), $l=1$ is the dipole, $l=2$ is the quadrupole, and so on. If you are outside a spherically symmetric object ($r>R$), you only need the monopole term, and the potential behaves as if all mass/charge were at the center (a foundational result). For non-symmetric objects, higher-order Legendre polynomials capture the shape's nuances. Solving boundary value problems, like the potential inside a spherical shell held at a specified voltage $V_0(\theta )$, directly employs Legendre series: you match the general series solution $V(r,\theta )={\sum }_l(A_l{r}^l+B_l{r}^{-l-1})P_l(\cos \theta )$ to the boundary condition using orthogonality to find the coefficients $A_l$ and $B_l$.

Common Pitfalls

1. Misapplying Orthogonality: The orthogonality condition ${\int }_{-1}^1{P}_m{P}_{n}dx=0$ only holds for the standard interval $[-1,1]$. If your physical problem is defined on a different interval (e.g., in angle $\theta$ from $0$ to $\pi$), you must perform a change of variable $x=\cos \theta$ and adjust the differential $dx=-\sin \theta d\theta$ correctly when computing coefficients.
2. Ignoring the Domain of Convergence: The Legendre series expansion of a function converges in the mean-square sense on $(-1,1)$. Like Fourier series, it may exhibit Gibbs phenomenon near discontinuities. It is not valid to blindly use the series outside the $x\in [-1,1]$ interval.
3. Confusing Normalization: Different texts occasionally use different normalization factors. The standard convention used here gives $P_l(1)=1$. Always check the normalization when combining formulas from different sources, especially in recurrence relations or integral identities.
4. Overlooking Physical Symmetry: Before launching into a full series solution, analyze the problem's symmetry. If the boundary condition is even in $x$, only even-$l$ polynomials will appear. This can halve your work and simplify calculations significantly.

Summary

• Legendre polynomials $P_l(x)$ are the polynomial solutions to Legendre's differential equation, arising naturally from solving Laplace's equation in spherical coordinates with azimuthal symmetry.
• They can be efficiently generated via the Rodrigues formula $P_l(x)=\frac{1}{2^{l}l!}\frac{d^l}{d{x}^l}(x^2-1{)}^l$ and are characterized by key properties like parity and specific recurrence relations.
• Their most powerful property is orthogonality on $[-1,1]$, which enables arbitrary functions on that interval to be expanded in a Legendre series, with coefficients determined by integration.
• The generating function $(1-2xt+t^2{)}^{-1/2}=\sum P_l(x)t^l$ provides a compact definition and directly yields the multipole expansion for the inverse distance $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$.
• Their primary application is in potential theory (gravitational and electrostatic), where they are indispensable for solving boundary value problems involving spherical geometries and for decomposing fields into monopole, dipole, and higher-order moments.