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Legendre Transformation

Table of Contents

1. Definition

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:

y=mx+b\begin{aligned}y = mx + b\end{aligned}

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

y=y′(x)x−b\begin{aligned}y = y'(x)x - b\end{aligned}

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

b=y′(x)x−y\begin{aligned}b = y'(x)x - y\end{aligned}

The actual Legendre Transform requires bb to be a function of y′y', therefore:

x(f′)=(f′(x))−1L{f(x)}=b(f′)=f′x(f′)−f((x(f′))\begin{aligned}x(f') = (f'(x))^{-1} \\L\{f(x)\} = b(f') = f'x(f') - f((x(f'))\end{aligned}

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