#+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*}