Subsection 3.1.2 Adrien-Marie Legendre

Years: 1752-1833

Citizenship: 🇮🇹🇫🇷 Italian-French

Core Contributions:

Developed the Legendre transform which converts convex functions into functions of their derivative variable.

The Legendre transform was later shown that it could be used as a computational tool to convert a system of \(n\) Euler-Lagrange Equations into a system of \(2n\) Hamiltonian equations.
Formulated Legendre’s differential equation.
Shows up in expansions for:
- Gravitational waves.
- Electric potentials when there is spherical symmetry.
  
  This bridges spherical harmonics and the wavefunction for the Hydrogen atom.
🔗
🔗
Derived the Legendre polynomials \(P_n\) as solutions to the Legendre equation.

Definition 3.1.1 Legendre’s Differential Equation.

\begin{equation*} (1-x^2)P_n''- xP_n' + n^2P_n \end{equation*}

\begin{equation*} P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \big(x^2 - 1\big)^n \end{equation*}

Singular points: \(\displaystyle x = \pm 1\)
Recurrence relation: \begin{equation*} (n+1)P_{n+1} = (2n+1)xP_n - nP_{n-1} \end{equation*}

where \(P_0 = 1\) and \(P_1 = x\text{.}\)