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:PROPERTIES:
:ID: 23df3ba6-ffb2-4805-b678-c5f167b681de
:END:
#+title: Legendre Transformation
#+author: Preston Pan
#+description: A tool used to translate between two pictures.
#+options: broken-links:t
* 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:
\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 [[id:83da205c-7966-417e-9b77-a0a354099f30][Lagrangian mechanics]], the Hamiltonian can be defined as the Legendre transform of the Lagrangian.
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