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\begin{document}

\newcommand\coursecode{VP}
\title{Variational Principles}
\author{}
\date{\today}
\maketitle

\tableofcontents

\newpage

\section{Variational Principles}

This is not intended to be a complete set of
notes. These notes only consist of the things that
I wanted to make flashcards of and practise
revising a little.

\subsection{Euler Lagrange Equations}

In this course, almost all of the problems will be
about minimising something like (where $x$ is a
function of $t$, so we call $F$ a
\emph{functional}):
\[ F[x] = \int_a^b f(x, \dot{x}, t) \dd t \]
In the case where $x$ must take predefined values
at $t = a$ and $t = b$, we can get \emph{Euler
Lagrange equations}.
The simplest form of Euler Lagrange:
\begin{flashcard}[Euler-Lagrange]
\prompt{Euler Lagrange equation?}
\[ \cloze{\pfrac{f}{x} = \dfrac{}{t} \left(
\pfrac{f}{\dot{x}} \right)} \]
\end{flashcard}
The derivation is to consider what we call the
\emph{first variation} of $F$:
\begin{align*}
    \delta F
    &= F[x + \delta x] - F[x] \\
    &= \int_a^b \left( f(x + \delta x, \dot{x} +
    \delta \dot{x}, t) - f(x, \dot{x}, t) \right)
    \dd t \\
    &= \int_a^b \left( \pfrac{f}{x} \delta x +
    \pfrac{f}{\dot{x}} \delta \dot{x} \right) \dd
    t \\
    &= \int_a^b \pfrac{f}{x} \delta x \dd t
    + \left[ \pfrac{f}{\dot{x}} \delta x \right]_a^b
    - \int_a^b \dfrac{}{t} \left( \pfrac{f}{\dot{x}}
    \right) \delta x \dd t \\
    &= \int_a^b \left( \pfrac{f}{x} - \dfrac{}{t}
    \left( \pfrac{f}{\dot{x}} \right) \right) \delta
    x \dd t
\end{align*}
Since $\delta x$ is arbitrary, we must have
\[ \pfrac{f}{x} - \dfrac{}{t} \left(
\pfrac{f}{\dot{x}} \right) = 0 \]

\subsubsection*{First Integrals}

There are also some useful first integrals of the
equation which are true under certain conditions
on $f$.

\myskip
\begin{flashcard}[Euler-Lagrange-no-x-dependence]
In the case where $f$ has no explicit $x$
dependence (i.e. $\pfrac{f}{x} = 0$), we get an
alternative form of Euler Lagrange:
\[ \cloze{\pfrac{f}{\dot{x}} = \text{const}} \]
\end{flashcard}

\myskip
In the case where $f$ has no explicit $t$
dependence (i.e. $\pfrac{f}{t} = 0$), we can also
get an alternative form of Euler Lagrange,
although the derivation is more involved.
\begin{flashcard}[Euler-Lagrange-no-t-dependence]
\prompt{Derivation of first integral for
$\pfrac{f}{t} = 0$?}
\cloze{
Consider $\dfrac{f}{t}$ using multivariate chain
rule:
\begin{align*}
    \dfrac{f}{t}
    &= \pfrac{t}{t} \ub{\pfrac{f}{t}}_{=0}
    + \pfrac{x}{t} \pfrac{f}{x}
    + \pfrac{\dot{x}}{t} \pfrac{f}{\dot{x}} \\
    &= \dot{x} \pfrac{f}{x}
    + \ddot{x} \pfrac{f}{\dot{x}}
\end{align*}
Multiplying the Euler Lagrange equation by $\dot{x}$
gives:
\begin{align*}
    0
    &= \dot{x} \pfrac{f}{x}
    - \dot{x} \dfrac{}{t} \left(
    \pfrac{f}{\dot{x}} \right) \\
    &= \dfrac{f}{t} - \ddot{x} \pfrac{f}{\dot{x}}
    - \dot{x} \dfrac{}{t} \left(
    \pfrac{f}{\dot{x}} \right) \\
    &= \dfrac{f}{t} - \dfrac{}{t} \left( \dot{x}
    \pfrac{f}{\dot{x}} \right) \\
    &= \dfrac{}{t} \left( f - \dot{x}
    \pfrac{f}{\dot{x}} \right)
\end{align*}
}
Hence we obtain the first integral:
\[ \boxed{\cloze{f - \dot{x} \pfrac{f}{\dot{x}} =
\text{const}}} \]
\end{flashcard}

\end{document}