Legendre Transformation

The Legendre Transformation represents a function in terms of the y-intercept of the tangent line at every point on the function. we start with the equation for a tangent line:

\begin{align*} y = mx + b \end{align*}

However, the Legendre transform actually solves for \(b\). For a general function \(f(x)\) we define the tangent line to a point on that function to be:

\begin{align*} y = y'(x)x - b \end{align*}

where subtracting \(b\) is the convention, for some reason. Then solving for b:

\begin{align*} b = y'(x)x - y \end{align*}

The actual Legendre Transform requires \(b\) to be a function of \(y'\), therefore:

\begin{align*} x(f') = (f'(x))^{-1} \\ L\{f(x)\} = b(f') = f'x(f') - f((x(f')) \end{align*}

In Lagrangian mechanics, the Hamiltonian can be defined as the Legendre transform of the Lagrangian.