Origin of Legendre's constant term.

I'm that student who needs to know where does something comes from. I have been studying Differential Equations and Electrodynamics (I'm a physics student), and I was wondering why we (in physics) use $$l(l+1)$$ as a constant in differential equations (and math...) which then result to be a Legendre's Differential Equation which can be solved through Legendre polynomials.

The equation is: $$\frac{1}{Sin(\theta )}\frac{d}{d\theta } \left( Sin(\theta ) \frac{d\Theta }{d\theta } \right) =-k\Theta$$ If we set (as the book suggests) the constant $$k=l(l+1)$$ (which is still a constant), the problem is simply solved through Legendre Polynomials. Why does that specific constant works? (As a general rule for Legendre's DE's/Polynomials).

Solving the problem gives... $$\frac{d}{d\theta } \left( Sin(\theta ) \frac{d\Theta }{d\theta } \right) =-l(l+1)Sin(\theta )\Theta$$ $$\Theta (\theta )=P_l[Cos(\theta )]$$ Thanks!

• Glancing through Wikipedia, it seems the rub is that you'll need to use an infinite series outside of the $k=\ell(\ell+1)$ case to solve that DE. In the particular case you're considering, the series happens to terminate in a polynomial. Perhaps there is some physical intuition behind this that I am unaware of. Feb 1, 2021 at 8:09

If the question is how do you get from your equation to the Legendre equation, from the context, I would expect you have $$\theta \in [0,\pi]$$. Then make the substitution $$\cos \theta = x$$, and the differential equation you have becomes, for $$x \in[-1,1]$$, \begin{align} (1-x^2) \frac {d^2 \Theta}{dx^2} - 2x \frac{d\Theta}{dx} + k\Theta = 0 \end{align} This is the Legendre equation. It has been studied extensively, and is described in the Wikipedia article referenced by Mr. Swanson in the comments. It has solutions known as the Legendre functions for general $$k$$, denoted $$P_\lambda$$ and $$Q_\lambda$$ where, by convention, $$k=\lambda(\lambda+1)$$ (I am using $$\lambda$$ instead of $$\ell$$ to match the article). The change of expression for $$k$$ is simply applied to turn your equation into a form that has recognized solutions.
When $$\lambda$$ is an integer, $$P_\lambda$$ a polynomial and the $$Q_\lambda$$ are known as a Legendre functions of the second kind. The boundary conditions for your equation will likely help determine which of these functions are relevant to your case. For example, applying boundary conditions may constrain $$k$$ to be an integer of the form $$\lambda(\lambda+1)$$ and you obtain the Legendre polynomials as an orthogonal family of solutions that forms a functional basis.