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diff --git a/mindmap/Legendre Transformation.org b/mindmap/Legendre Transformation.org new file mode 100644 index 0000000..f9bc51f --- /dev/null +++ b/mindmap/Legendre Transformation.org @@ -0,0 +1,31 @@ +:PROPERTIES: +:ID: 23df3ba6-ffb2-4805-b678-c5f167b681de +:END: +#+title: Legendre Transformation +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+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. |