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+#+title:Variation Calculus
+#+author: Thomas Albers Raviola
+#+date: 2022-10-01
+#+setupfile: ../../math_options.org
+
+* Disclaimer
+This site as of now just a technology demonstration and its claims
+should not be taken as true (even though I myself am pretty confident
+they are)
+
+* Beltrami identity
+\begin{align*}
+\deriv{}{x}\pderiv{F}{\dot{y}} &= \pderiv{F}{y}\\
+\deriv{}{x}\left(\pderiv{F}{\dot{y}}\right)\deriv{y}{x} &= \pderiv{F}{y}\deriv{y}{x}\\
+\deriv{}{x}\left(\pderiv{F}{\dot{y}}\right)\deriv{y}{x} + \pderiv{F}{\dot{y}}\deriv{\dot{y}}{x} + \pderiv{F}{x} &= \pderiv{F}{y}\deriv{y}{x} + \pderiv{F}{\dot{y}} \deriv{\dot{y}}{x} + \pderiv{F}{x}\\
+\deriv{}{x}\left(\pderiv{F}{\dot{y}}\dot{y}\right) + \pderiv{F}{x} &= \deriv{F}{x}\\
+\deriv{}{x}\left(\pderiv{F}{\dot{y}}\dot{y} - F\right) &= - \pderiv{F}{x}\\
+\pderiv{F}{\dot{y}}\dot{y} - F\right &= C
+\end{align*}