fem_maxwellian.lyx

#LyX 2.1 created this file. For more info see http://www.lyx.org/
\lyxformat 474
\begin_document
\begin_header
\textclass revtex4-1
\use_default_options true
\maintain_unincluded_children false
\language british
\language_package default
\inputencoding auto
\fontencoding global
\font_roman default
\font_sans default
\font_typewriter default
\font_math auto
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation skip
\defskip smallskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\non}{}
\end_inset


\begin_inset FormulaMacro
\newcommand{\difp}[2]{\frac{\partial#1}{\partial#2}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\br}{{\bf r}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bR}{{\bf R}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bA}{{\bf A}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bB}{{\bf B}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bE}{{\bf E}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bm}{{\bf m}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bn}{{\bf n}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bN}{{\bf N}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bp}{{\bf p}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bF}{{\bf F}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bz}{{\bf z}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bZ}{{\bf Z}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bV}{{\bf V}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\bv}{{\bf v}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bu}{{\bf u}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bx}{{\bf x}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bX}{{\bf X}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bJ}{{\bf J}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bj}{{\bf j}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bk}{{\bf k}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bTheta}{{\bf \Theta}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\btheta}{\boldsymbol{\theta}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bOmega}{\boldsymbol{\Omega}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bomega}{\boldsymbol{\omega}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\brho}{\boldsymbol{\rho}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\rd}{{\rm d}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\rJ}{{\rm J}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ph}{\varphi}
\end_inset


\begin_inset FormulaMacro
\newcommand{\te}{\theta}
\end_inset


\begin_inset FormulaMacro
\newcommand{\tht}{\vartheta}
\end_inset


\begin_inset FormulaMacro
\newcommand{\vparkb}{v_{\parallel k b}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\vparkm}{v_{\parallel k m}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\Jpar}{J_{\parallel}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ppar}{p_{\parallel}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\Bpstar}{B_{\parallel}^{*}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\intpi}{\int\limits _{0}^{2\pi}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\summ}{\sum\limits _{m=-\infty}^{\infty}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\tb}{\tau_{b}(\uv)}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bh}{{\bf h}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\cE}{{\cal E}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\odtwo}[2]{\frac{\rd#1}{\rd#2}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\pdone}[1]{\frac{\partial}{\partial#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\pdtwo}[2]{\frac{\partial#1}{\partial#2}}
{\frac{\partial#1}{\partial#2}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ds}{{\displaystyle }}
\end_inset


\begin_inset FormulaMacro
\newcommand{\iotabar}{\mbox{\ensuremath{\iota\!\!}-}}
{\tilde{\iota}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\vpar}{v_{\parallel}}
\end_inset


\end_layout

\begin_layout Section
Electromagnetic Field Equations
\end_layout

\begin_layout Standard
The aim of this write-up is to summarise numerical formulations for the
 solution of Maxwell's equations
\begin_inset Formula 
\begin{eqnarray}
{\rm div}\mathbf{E} & = & 4\pi\rho,\label{eq:divE}\\
{\rm curl}\mathbf{E} & = & -\frac{1}{c}\difp{\bB}t,\label{eq:rotE}\\
{\rm curl}\mathbf{B} & = & \frac{4\pi}{c}\bJ+\frac{1}{c}\difp{\bE}t,\label{eq:rotB}\\
{\rm div}\mathbf{B} & = & 0.\label{eq:divB}
\end{eqnarray}

\end_inset

The continuity equation 
\begin_inset Formula 
\begin{equation}
{\rm div}\bJ+\difp{\rho}t=0\label{eq:cont}
\end{equation}

\end_inset

for charges and currents follows from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:divE"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rotB"

\end_inset

.
\end_layout

\begin_layout Standard
To trivially fulfill and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rotE"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:divB"

\end_inset

 due to vector identities it is common to use the potential formulations
\begin_inset Formula 
\begin{eqnarray}
\bE & = & -\nabla\Phi-\frac{1}{c}\difp{\mathbf{A}}t,\\
\bB & = & {\rm curl}\bA.
\end{eqnarray}

\end_inset

The equations for the potentials from the remaining Maxwell equations are
\begin_inset Formula 
\begin{eqnarray}
-\Delta\Phi-\frac{1}{c}\pdone t{\rm div}\mathbf{A} & = & 4\pi\rho,\label{eq:pot1}\\
\left({\rm curl}\,{\rm {\rm curl}}+\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\right)\mathbf{A}+\frac{1}{c}\pdone t\nabla\Phi & = & \frac{4\pi}{c}\mathbf{J}.\label{eq:pot2}
\end{eqnarray}

\end_inset

Their solution is unique up to an arbitrary gauge transformation
\begin_inset Formula 
\begin{eqnarray}
\Phi^{\prime} & = & \Phi-\frac{1}{c}\difp{\chi}t,\\
\mathbf{A}^{\prime} & = & \mathbf{A}+\nabla\chi
\end{eqnarray}

\end_inset

with gauge function 
\begin_inset Formula $\chi$
\end_inset

.
 It is possible to decouple the potential equations by choice of the Lorentz
 gauge.
\end_layout

\begin_layout Subsection
Time-harmonic case
\end_layout

\begin_layout Standard
By performing a Fourier transformation
\begin_inset Formula 
\begin{align*}
f(t) & =\frac{1}{\sqrt{2\pi}}\int d\omega f(\omega)e^{-i\omega t}
\end{align*}

\end_inset

for all functions one can write the basic equations as
\begin_inset Formula 
\begin{eqnarray}
{\rm div}\mathbf{E} & = & 4\pi\rho,\label{eq:divE-1}\\
{\rm curl}\mathbf{E} & = & \frac{i\omega}{c}\bB,\label{eq:rotE-1}\\
{\rm curl}\mathbf{B} & = & \frac{4\pi}{c}\bJ-\frac{i\omega}{c}\bE,\label{eq:rotB-1}\\
{\rm div}\mathbf{B} & = & 0,\label{eq:divB-1}
\end{eqnarray}

\end_inset

the continuity equation as
\begin_inset Formula 
\begin{equation}
{\rm div}\bJ-i\omega\rho=0,\label{eq:cont-1}
\end{equation}

\end_inset

and the potential equations as
\begin_inset Formula 
\begin{eqnarray}
-\Delta\Phi+\frac{i\omega}{c}{\rm div}\mathbf{A} & = & 4\pi\rho,\label{eq:pot1-1}\\
\left({\rm curl}\,{\rm {\rm curl}}-\frac{\omega^{2}}{c^{2}}\right)\mathbf{A}-\frac{i\omega}{c}\nabla\Phi & = & \frac{4\pi}{c}\mathbf{J}.\label{eq:pot2-1}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Subsection
Static case
\end_layout

\begin_layout Standard
For the static case 
\begin_inset Formula $\omega\rightarrow0$
\end_inset

 one can write the basic equations as
\begin_inset Formula 
\begin{eqnarray}
{\rm div}\mathbf{E} & = & 4\pi\rho,\label{eq:divE-1-1}\\
{\rm curl}\mathbf{E} & = & 0,\label{eq:rotE-1-1}\\
{\rm curl}\mathbf{B} & = & \frac{4\pi}{c}\bJ,\label{eq:rotB-1-1}\\
{\rm div}\mathbf{B} & = & 0,\label{eq:divB-1-1}
\end{eqnarray}

\end_inset

the continuity equation as
\begin_inset Formula 
\begin{equation}
{\rm div}\bJ=0,\label{eq:cont-1-1}
\end{equation}

\end_inset

and the potential equations as
\begin_inset Formula 
\begin{eqnarray}
-\Delta\Phi & = & 4\pi\rho,\label{eq:pot1-1-1}\\
{\rm curl}\,{\rm {\rm curl}}\mathbf{A} & = & \frac{4\pi}{c}\mathbf{J}.\label{eq:pot2-1-1}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Subsubsection
Integrals for Coulomb's equation and Biot Savart law
\end_layout

\begin_layout Standard
For open domains, the electrostatic field fulfilling the divergence equation
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:divE-1-1"

\end_inset

 can be written as a volume integral over the electric charge density by
 Coulomb's equation
\begin_inset Formula 
\begin{align}
\mathbf{E}(\mathbf{x}) & =\int d^{3}x^{\prime}\rho(\mathbf{x}^{\prime})\frac{(\mathbf{x}-\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}.
\end{align}

\end_inset

The magnetostatic field fulfilling Ampère's law 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rotB-1-1"

\end_inset

 can be written as a volume integral over the current density by the Biot
 Savart law
\begin_inset Formula 
\begin{align}
\mathbf{B}(\mathbf{x}) & =\frac{1}{c}\int d^{3}x^{\prime}\mathbf{J}(\mathbf{x}^{\prime})\times\frac{(\mathbf{x}-\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}\label{eq:BiotSavart}
\end{align}

\end_inset

in the free field.
 Using the relation
\begin_inset Formula 
\begin{align}
\frac{(\mathbf{x}-\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}} & =-\nabla\frac{1}{|\mathbf{x}-\mathbf{x}^{\prime}|}
\end{align}

\end_inset

we can write the two laws for the free field potentials in Coulomb gauge
 as
\begin_inset Formula 
\begin{align}
\Phi(\mathbf{x}) & =\int d^{3}x^{\prime}\frac{\rho(\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|},\\
\mathbf{A}(\mathbf{x}) & =\frac{1}{c}\nabla\times\int d^{3}x^{\prime}\frac{\mathbf{J}(\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|}.
\end{align}

\end_inset


\end_layout

\begin_layout Subsubsection
Stream function
\end_layout

\begin_layout Standard
In an arbitrary (also finite) domain 
\begin_inset Formula $\Omega$
\end_inset

, the divergence-free current can be represented by 
\begin_inset Formula 
\begin{equation}
\mathbf{J}=\nabla\times\mathbf{T}
\end{equation}

\end_inset

via the stream function
\begin_inset Formula 
\begin{align}
\mathbf{T} & =\frac{1}{4\pi}\int_{\Omega}d^{3}x^{\prime}\mathbf{J}(\mathbf{x}^{\prime})\times\frac{(\mathbf{x}-\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}.
\end{align}

\end_inset

For the open space 
\begin_inset Formula $\Omega=\mathbb{R}^{3}$
\end_inset

, we can identify
\begin_inset Formula 
\begin{align}
\mathbf{B} & =\frac{4\pi}{c}\mathbf{T}
\end{align}

\end_inset

by comparison to the Biot Savart law 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:BiotSavart"

\end_inset

.
 In the most general case from Ampere's law 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rotB-1-1"

\end_inset

, the magnetic field is given by
\begin_inset Formula 
\begin{align}
\mathbf{B} & =\frac{4\pi}{c}(\mathbf{T}-\nabla\Phi_{m}).\label{eq:BT}
\end{align}

\end_inset

Eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:BT"

\end_inset

 may be interpreted as a superposition of two solutions: 
\begin_inset Formula $\mathbf{T}$
\end_inset

 represents the contributions from the inhomogenous equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rotB-1-1"

\end_inset

 in 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 and 
\begin_inset Formula $\nabla\Phi_{m}$
\end_inset

 the ones from the boundary value problem of the current-free equation in
 the finite domain 
\begin_inset Formula $\Omega$
\end_inset

.
 To make 
\begin_inset Formula $\mathbf{B}$
\end_inset

 divergence free, the magnetic scalar potential 
\begin_inset Formula $\Phi_{m}$
\end_inset

 has to fullfil Poisson's equation
\begin_inset Formula 
\begin{align}
\Delta\Phi_{m} & ={\rm div}\mathbf{T}\label{eq:phim}
\end{align}

\end_inset

in 
\begin_inset Formula $\Omega$
\end_inset

.
\end_layout

\begin_layout Section
Boundary value problems
\end_layout

\begin_layout Subsection
Magnetostatics
\end_layout

\begin_layout Standard
In the vector potential formulation by 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:pot2-1-1"

\end_inset

, two specific kinds of boundary values are
\end_layout

\begin_layout Enumerate
Dirichlet boundary values for 
\begin_inset Formula $\mathbf{A}$
\end_inset

,
\begin_inset Formula 
\begin{align}
\mathbf{n}\times\mathbf{A} & =\boldsymbol{\alpha},
\end{align}

\end_inset

where taking the divergence yields
\begin_inset Formula 
\begin{align}
{\rm div}(\mathbf{n}\times\mathbf{A}) & =\mathbf{n}\cdot{\rm curl}\mathbf{A}-\mathbf{A}\cdot\underbrace{{\rm curl}\mathbf{n}}_{=0}\nonumber \\
\Rightarrow\mathbf{n}\cdot\mathbf{B} & ={\rm div}\boldsymbol{\alpha}=:b,\label{eq:diva}
\end{align}

\end_inset

and
\end_layout

\begin_layout Enumerate
Neumann boundary values for 
\begin_inset Formula $\mathbf{A}$
\end_inset

,
\begin_inset Formula 
\begin{align}
\mathbf{n}\times{\rm curl}\mathbf{A} & =\mathbf{K}=\mathbf{n}\times\mathbf{B}.\label{eq:ncurlA}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
In the scalar potential formulation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:phim"

\end_inset

 the two types of boundary conditions swap their role with
\end_layout

\begin_layout Enumerate
Dirichlet boundary values for 
\begin_inset Formula $\Phi_{m}$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\Phi_{m}=\Phi_{m0},
\end{equation}

\end_inset

which is equivalent to 
\begin_inset Formula 
\begin{align}
\mathbf{n}\times\mathbf{B} & =\mathbf{K}
\end{align}

\end_inset

TODO: explain why, and
\end_layout

\begin_layout Enumerate
Neumann boundary values for 
\begin_inset Formula $\Phi_{m}$
\end_inset

,
\begin_inset Formula 
\begin{align}
\frac{\partial\Phi_{m}}{\partial n} & =b=\mathbf{n}\cdot\mathbf{B}.\label{eq:b2}
\end{align}

\end_inset


\end_layout

\begin_layout Section
Open boundary conditions
\end_layout

\begin_layout Subsection
Vector potential
\end_layout

\begin_layout Standard
Let us consider the problem in 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

: 
\begin_inset Formula 
\begin{align}
{\rm curl}\,{\rm {\rm curl}}\mathbf{A} & =\frac{4\pi}{c}\mathbf{J},\\
\mathbf{A} & {\rm decaying\, appropriately\, at\,\mbox{\infty}.}
\end{align}

\end_inset

TODO: more accurate formulation.
 We formulate an equivalent coupled problem in the bounded domain 
\begin_inset Formula $\Omega$
\end_inset

 with a magnetic scalar potential formulation in a current free exterior
 domain,
\begin_inset Formula 
\begin{align}
{\rm curl}\,{\rm {\rm curl}}\mathbf{A} & =\frac{4\pi}{c}\mathbf{J}\,{\rm in\,\Omega},\\
\Delta\Phi_{m} & =0\,{\rm in}\,\mathbb{R}^{3}\backslash\Omega,\\
{\rm compatibility} & {\rm between\,\Phi\, and\,\mathbf{A}\, on\,\Gamma}\\
\Phi_{m} & {\rm decaying\, appropriately\, at\,\mbox{\infty}.}
\end{align}

\end_inset

TODO: formulate more accurately.
 The integral current must vanish inside 
\begin_inset Formula $\Omega$
\end_inset

 for the compatibility condition.
 Let us assume the Neumann problem 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:b2"

\end_inset

 to be solved on the exterior domain by the linear operator (Neumann-to-Dirichle
t-map) 
\begin_inset Formula $L$
\end_inset

 acting on the boundary values,
\begin_inset Formula 
\begin{align}
\Phi_{m} & =Lb\\
\Rightarrow\mathbf{B} & ={\rm grad}\, Lb.
\end{align}

\end_inset

Now we express 
\begin_inset Formula $b$
\end_inset

 by 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:diva"

\end_inset

 and insert the expression into 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:ncurlA"

\end_inset

:
\begin_inset Formula 
\begin{align}
\mathbf{n}\times{\rm curl}\mathbf{A} & =\mathbf{K}\nonumber \\
 & =\mathbf{n}\times\mathbf{B}\nonumber \\
 & =\mathbf{n}\times{\rm grad}\, Lb\nonumber \\
 & =\mathbf{n}\times{\rm grad}\, L({\rm div}(\mathbf{n}\times\mathbf{A})).
\end{align}

\end_inset

Thus, we have reduced the exterior equations to a relation between Dirichlet
 and Neumann values for 
\begin_inset Formula $\mathbf{A}$
\end_inset

 and write the equivalent problem as
\begin_inset Formula 
\begin{align}
{\rm curl}\,{\rm {\rm curl}}\mathbf{A} & =\frac{4\pi}{c}\mathbf{J}\,{\rm in\,\Omega},\\
\mathbf{n}\times{\rm curl}\mathbf{A} & =M(\mathbf{n}\times\mathbf{A})\,{\rm on\,\Gamma},
\end{align}

\end_inset

where
\begin_inset Formula 
\begin{align}
M\mathbf{u} & =\mathbf{n}\times{\rm grad}\, L({\rm div}\mathbf{u}).
\end{align}

\end_inset

TODO: depending on the form of 
\begin_inset Formula $L$
\end_inset

 this might be simplified by
\begin_inset Formula 
\begin{align}
{\rm curl}(f\mathbf{v}) & =f{\rm curl}\mathbf{v}+{\rm grad}f\times\mathbf{v}.
\end{align}

\end_inset


\end_layout

\begin_layout Section
Variational form and Finite Elements
\end_layout

\begin_layout Standard
For numerical treatment it is convenient to write the equivalent variational
 problem to a partial differential equation.
 It is the opposite strategy to the one used in mechanics where the equations
 of motion are derived from a variational principle and solved in a differential
 formulation.
 In Lagrangian mechanics, the three equivalent formulations were
\end_layout

\begin_layout Enumerate
Stationary action: The actual motion 
\begin_inset Formula $\mathbf{q}(t)$
\end_inset

 of a mechanical system makes the action
\begin_inset Formula 
\begin{align}
S[\mathbf{q}(t)] & =\int_{t_{0}}^{t_{1}}L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)dt
\end{align}

\end_inset

stationary (a maximum, minimum or saddle point), where the Lagrangian 
\begin_inset Formula $L$
\end_inset

 is given by the difference between kinetic and potential energy.
 All paths 
\begin_inset Formula $\mathbf{q}(t)$
\end_inset

 that start at the previously defined 
\begin_inset Formula $\mathbf{q}(t_{1})$
\end_inset

 and end at 
\begin_inset Formula $\mathbf{q}(t_{2})$
\end_inset

 are taken into account.
 Since 
\begin_inset Formula $S$
\end_inset

 assigns a scalar value to each possible path 
\begin_inset Formula $\mathbf{q}(t)$
\end_inset

, it is also called the action functional.
\end_layout

\begin_layout Enumerate
Vanishing first variation: The first variation of the action 
\begin_inset Formula $\delta S[\mathbf{q}(t)]$
\end_inset

 must vanish.
\begin_inset Newline newline
\end_inset

We assume that a solution 
\begin_inset Formula $\mathbf{q}(t)$
\end_inset

 exists and develop the path around this solution by adding a small variation
\begin_inset Formula 
\begin{align}
\delta\mathbf{q}(t) & =\varepsilon\boldsymbol{\eta}(t).
\end{align}

\end_inset


\begin_inset Formula $\varepsilon$
\end_inset

 is a constant and the function 
\begin_inset Formula $\boldsymbol{\eta}(t)$
\end_inset

 must vanish at the endpoints.
 We notice that 
\begin_inset Formula 
\begin{align}
\frac{d}{dt}\left(\mathbf{q}(t)+\delta\mathbf{q}(t)\right) & =\dot{\mathbf{q}}(t)+\varepsilon\dot{\boldsymbol{\eta}}(t)=\dot{\mathbf{q}}(t)+\delta\dot{\mathbf{q}}(t).
\end{align}

\end_inset

For small 
\begin_inset Formula $\varepsilon$
\end_inset

, the variation of the action around the solution 
\begin_inset Formula $\mathbf{q}(t)$
\end_inset

 is found by Taylor expansion of the Lagrangian,
\begin_inset Formula 
\begin{align}
S[\mathbf{q}(t)+\varepsilon\boldsymbol{\eta}(t)]-S[\mathbf{q}(t)] & =\int_{t_{0}}^{t_{1}}L(\mathbf{q}(t)+\varepsilon\boldsymbol{\eta}(t),\dot{\mathbf{q}}(t)+\varepsilon\dot{\boldsymbol{\eta}}(t),t)dt-S[\mathbf{q}(t)]\nonumber \\
 & =\varepsilon\int_{t_{0}}^{t_{1}}\left(\boldsymbol{\eta}\cdot\frac{\partial L}{\partial\mathbf{q}}+\dot{\boldsymbol{\eta}}\cdot\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)dt+\mathcal{O}(\varepsilon^{2})\overset{!}{=}0.
\end{align}

\end_inset

For arbitrarily small 
\begin_inset Formula $\varepsilon$
\end_inset

, the first term of linear order (called the first variation 
\begin_inset Formula $\delta S[\mathbf{q}(t)]$
\end_inset

 of 
\begin_inset Formula $S[\mathbf{q}(t)]$
\end_inset

) dominates and must thus vanish individually.
 Integration by parts (
\begin_inset Formula $\boldsymbol{\eta}(t_{1})=\boldsymbol{\eta}(t_{2})=0$
\end_inset

) yields
\begin_inset Formula 
\begin{align}
\delta S[\mathbf{q}(t)] & =\varepsilon\int_{t_{0}}^{t_{1}}\left(\boldsymbol{\eta}\cdot\frac{\partial L}{\partial\mathbf{q}}-\boldsymbol{\eta}\cdot\frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)dt\nonumber \\
 & =\int_{t_{0}}^{t_{1}}\delta\mathbf{q}\cdot\left(\frac{\partial L}{\partial\mathbf{q}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)dt\overset{!}{=}0.
\end{align}

\end_inset


\end_layout

\begin_layout Enumerate
Euler-Lagrange differential equations: Since we can choose any function
 
\begin_inset Formula $\boldsymbol{\eta}(t)$
\end_inset

 around the solution, we require the term inside the brackets to vanish,
\begin_inset Formula 
\begin{align}
\frac{\partial L}{\partial\mathbf{q}}-\frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}} & =\mathbf{0}.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
For partial differential equations, the principle is the same, but with
 more than one variable.
 We have
\end_layout

\begin_layout Enumerate
Stationary energy functional:
\end_layout

\end_body
\end_document