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Legendre's Equation Model
Shodor > CSERD > Resources > Models > Legendre's Equation Model
  Software  •  Instructions  •  Theory

Instructions - Legendre's Equation Model

Purpose

This applet numerically solves the Legendre equation


\begin{displaymath} - \left[ \frac{d}{d x} \left( (1-x^2) \frac{d }{d x} \right) +\frac{m^2}{(1-x^2)} \right] P(x) = \lambda P(x) \end{displaymath}

The solution to this equation are the Associated Legendre Functions, the most commonly used of which are the Legendre Polynomials, which are the $m=0$ solutions to the equation.

The equation has singularities at $x = \pm 1$ and solutions in the range $-1 \leq x \geq 1$ only exist for integer values of m. In addition, for any given value of m, the eigenvalues of the equation are found to be $l (l+1)$, and only exist for $l \geq \vert m\vert$.

Fundamentals

The solution is solved from near -1 to near 1. The graph displays backwards because the modeling environment used does not allow for an integration in which the independent variable is decreasing, and it was desired to specify the boundary conditions such that $P(1) \geq 0$.

You can modify the parameters $\lambda$ and $m$ by changing the sliderbars.

From the form of the equation we can require the following of a physically meaningful solution to the equation. First, the equation has the potential for singularities at x = +- 1, so a physically meaningful solution should be finite at the boundaries.

For m>0, it is clear that this will only be true if P(-1) = P(1) =0.

Making the substitution U = d P/d x one can turn the Legendre equation for m>0 into a system of equations, and require that both U and P be finite and differentiable in the range x = [ -1, 1].

For m=0, it is a little more difficult to see what the boundary conditions should be. However, if the substitution U = (1-x2) d P/ d x is made into the above equation, one can get a system of equations for which the boundary conditions U(-1) = U(1) = 0 must hold.

The tool solves for U and P with U = (1-x2) d P/d x for m=0, and U = d P/d x for m>0.

Things to Try

Do you find that the values of $\lambda$ and $m$ from the numerical solution are in agreement with the standard eigenvalues described for this equation?

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