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:

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:

where subtracting $b$ is the convention, for some reason. Then solving for b:

The actual Legendre Transform requires $b$ to be a function of $y^{\prime }$, therefore:

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