electromagnetism - What is the physical meaning/concept behind Legendre polynomials?

In mathematical physics and other textbooks we find the Legendre polynomials are solutions of Legendre's differential equations. But I didn't understand where we encounter Legendre's differential equations (physical example). What is the basic physical concept behind the Legendre polynomials? How important are they in physics? Please explain simply and give a physical example.

Answer

The Legendre polynomials occur whenever you solve a differential equation containing the Laplace operator in spherical coordinates with a separation ansatz (there is extensive literature on all of those keywords on the internet).

Since the Laplace operator appears in many important equations (wave equation, Schrödinger equation, electrostatics, heat conductance), the Legendre polynomials are used all over physics.

There is no (inarguable) physical concept behind the Legendre polynomials, they are just mathematical objects which form a complete basis between -1 and 1 (as do the Chebyshev polynomials).