\documentclass[main.tex]{subfiles}

\begin{document}
\subsection{2D problem in cartesian coordinates: horizontal launch}
\SlideWithFullFigureWithLink{Experiment setup \\(credit:\href{https://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontal-and-Vertical-Components-of-Velocity}{The Physics clasroom})}{examples/Spreadsheets/2D_HorizontalLaunch_Basic/PhysClass.png}{https://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontal-and-Vertical-Components-of-Velocity}
\SlideWithFullFigureWithLink{Spreadsheet approach}{examples/Spreadsheets/2D_HorizontalLaunch_Basic/Screenshot.png}{http://buon.fjfi.cvut.cz:5002/CTU4WP@ELAB/examples/Spreadsheets/2D_HorizontalLaunch_Basic/}
\SlideWithFullFigureWithLink{Processing approach}{examples/Processing/HorizontalLaunch/Screenshot.png}{https://editor.p5js.org/vojtech.svob/sketches/3O2OUtpW}

\subsection{Runge Kutta}
\slide{Runge Kutta method}{
\blok{Let an initial value problem be specified:}
{\centering $\dot {y}=f(t,y),\quad y(t_{0})=y_{0}$}
\twocolumns{0.5}{
\fig{1}{450px-Runge-Kutta_slopes.png}{Slopes used by the classical Runge-Kutta method \cite{wiki:RK}}{450px-Runge-Kutta_slopes}
}{0.5}{
\begin{align*}
y_{n+1} &= y_n + \tfrac{1}{6}\left(k_1 + 2k_2 + 2k_3 + k_4 \right),\\
t_{n+1} &= t_n + h \\
\end{align*}
\begin{align*}
 k_1 &= h\ f(t_n, y_n), \\
 k_2 &= h\ f\left(t_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right), \\ 
 k_3 &= h\ f\left(t_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right), \\
 k_4 &= h\ f\left(t_n + h, y_n + k_3\right).
\end{align*}
}
}

\slide{Runge-Kutta versus Euler method}{
\fig{0.7}{Runge-kutta}{Runge-Kutta methods for the differential equation $y'=sin(t)^2\cdot y$ \cite{wiki:RK}}{wiki:RK}
}


\subsection{ODE solving with standard functions}

\SlideWithFullFigureWithLink{odeint: Python solver}{examples/PythonJupyterNotebook/Pendulum_odeint/Screenshot.png}{http://buon.fjfi.cvut.cz:5002/CTU4WP@ELAB/examples/PythonJupyterNotebook/Pendulum_odeint/Screenshot.png}


\subsection{Foucalt pendulum}
\slide{Foucalt pendulum}{
\fig{0.4}{768pxPantheonPenduledeFoucault2.jpeg}{\cite{wiki:Foucalt}}{768pxPantheonPenduledeFoucault2.jpg}
}

\slide{Foucault pendulum - dynamic equations }{
\twocolumns{0.35}{
\fig{1}{Foucalt.jpeg}{Foucault pendulum - setup}{}
}{0.65}{
\centering
Coriolis force:
\begin{align*}
F_{c,x} &=2 m \Omega \dfrac{dy}{dt} \sin\varphi\\
F_{c,y} &=- 2 m \Omega \dfrac{dx}{dt} \sin\varphi
\end{align*}
Restoring force (small angle approximation):
\begin{align*}
F_{g,x} &=- m \omega^2 x \\
F_{g,y} &=- m \omega^2 y.
\end{align*}
Then dynamic equations:
\begin{align*}
\dfrac{d^2x}{dt^2} &=-\omega^2 x + 2 \Omega \dfrac{dy}{dt} \sin \varphi\\
\dfrac{d^2y}{dt^2} &=-\omega^2 y - 2 \Omega \dfrac{dx}{dt} \sin \varphi .
\end{align*}
}
}

\SlideWithFullFigureWithLink{Foucault pendulum @ processing}{examples/Processing/FoucaultPendulum/Screenshot.png}{https://editor.p5js.org/vojtech.svob/sketches/DMgIN9Cq}

\subsection{Satellite motion}
\SlideWithFullFigureWithLink{Satellite motion @ processing}{examples/Processing/SatelliteMotion/Screenshot.png}{https://editor.p5js.org/vojtech.svob/sketches/m57euT3SL}


\end{document}
