\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{De Donder-Weyl-Hamilton equation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{variational_calculus}{}\paragraph*{{Variational calculus}}\label{variational_calculus}

[[!include variational calculus - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsnplecticHamiltonEquation}{As a multisymplectic/$n$-plectic Hamiltonian equation}\dotfill \pageref*{AsnplecticHamiltonEquation} \linebreak
\noindent\hyperlink{relativistic_form}{Relativistic form}\dotfill \pageref*{relativistic_form} \linebreak
\noindent\hyperlink{nonrelativistic_form}{Non-relativistic form}\dotfill \pageref*{nonrelativistic_form} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{De Donder-Weyl-Hamilton equation} is a refinement of [[Hamilton's equation]] from a single [[variational derivative|variational]] direction to many.

Where [[Hamilton's equation]] appears as a natural condition in [[symplectic geometry]], so the De Donder-Weyl-Hamilton equation appears as the analogous condition in [[multisymplectic geometry]]/[[n-plectic geometry]].

Where [[Hamilton's equation]] appears as the [[equations of motion]] of a [[mechanical system]] whose [[dynamics]] is described by evolution along a single parameter ([[time]]), so the \emph{De Donder-Weyl Hamilton equation} appears as the [[equations of motion]] of a higher dimensional [[field theory]] given by a [[local Lagrangian]] where ``evolution'' is along more parameters ([[spacetime]]).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\Sigma$ be a [[smooth manifold]] to be regarded as [[worldvolume]]/[[spacetime]] and let $F \to \Sigma$ be a smooth [[bundle]] to be thought of as the [[field bundle]].

For simplicity of exposition we first consider local patches and take without essential restriction $\Sigma$ to be a [[Cartesian space]] and the [[field bundle]] to be a trivial [[vector bundle]] $X \times \Sigma$. In all of the following the [[summation convention]] for summation over repeated induces is understood.

We then denote the canonical [[coordinates]] on $\Sigma$ by $\{\sigma^\mu\}$ and those on $X$ by $\{\phi^i\}$.

Write $(J^1 E^)^\ast$ for the dualized first [[jet bundle]] of $E$. Under the above assumptions this has canonical local coordinates $\{\sigma^\mu, \phi^i, \pi_i^\mu\}$. Here we call $\pi^\mu_i$ the ``$\mu$-th [[momentum]] of the $i$-th field''.

Consider a [[local Lagrangian]] $L$ which is \emph{of first order}, hence which is a function on the first [[jet bundle]] $J^1 E$ of the [[field bundle]].

\begin{defn}
\label{ELEquations}\hypertarget{ELEquations}{}
Its [[Euler-Lagrange equations]] are

\begin{displaymath}
\partial_\mu
  \frac{\delta L }{\delta (\partial_\mu \phi^i)} 
   = 
  \frac{\delta L}{ \delta \phi^i}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{dWHamiltonianByLegendre}\hypertarget{dWHamiltonianByLegendre}{}
The \emph{De Donder-Weyl-Hamiltonian} of $L$ is its generalized [[Legendre transform]] (if it exists), the function

\begin{displaymath}
H \coloneqq \pi^\mu_i \partial_\mu \phi^i - L
\end{displaymath}
on the dual jet bundle.

\end{defn}
\begin{prop}
\label{DWHEquations}\hypertarget{DWHEquations}{}
In terms of the De Donder-Weyl Hamiltonian $H$, def. \ref{dWHamiltonianByLegendre}, the [[Euler-Lagrange equations of motion]], def. \ref{ELEquations}, are equivalent to the [[differential equations]]

\begin{displaymath}
\partial_\mu \phi^i
    =
   \frac{\partial H}{\partial \pi_i^\mu} 
  \;\;\,,
  \;\;\;\;
  \partial_\mu \pi^\mu_i
   =
  -
  \frac{\partial H}{ \partial \phi^i} 
  \,.
\end{displaymath}
These are called the \textbf{De Donder-Weyl-Hamilton equations}.

\end{prop}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AsnplecticHamiltonEquation}{}\subsubsection*{{As a multisymplectic/$n$-plectic Hamiltonian equation}}\label{AsnplecticHamiltonEquation}

For $(X,\omega)$ a [[symplectic manifold]] and $H \in C^\infty(X)$ a [[smooth function]] regarded as a [[Hamiltonian]], then [[Hamilton's equations]] are equivalent to

\begin{displaymath}
\iota_v \omega = \mathbf{d}H
\end{displaymath}
for $v$ a [[tangent vector]] to a [[trajectory]] in [[phase space]]. This may be referred to as the ``non-relativistic'' form of the symplectic version of Hamilton's equations, as the ``[[time]]''-parameter is not part of the [[phase space]].

By passing from plain phase space to the corresponding [[multisymplectic geometry|dual jet bundle]], hence by adjoining a [[worldline]] coordinate $t$, this is equivalent to

\begin{displaymath}
\iota_v \Omega = 0
\end{displaymath}
where now

\begin{displaymath}
\Omega = \omega + \mathbf{d}H \wedge \mathbf{d}t
  \,.
\end{displaymath}
This may accordingly be thought of as the [[relativity|relativistic]] version of Hamilton's equations.

We now discuss the analog of both the ``non-relativistic'' and of this ``[[relativity|relativistic]] version'' of [[Hamilton's equation]] for the de Donder-Weyl-Hamilton equation.

In both cases, the $n$-tuple of [[tangent vectors]] to a [[section]] which satisfies the [[equations of motion]] is characterized as a \emph{[[Hamiltonian n-vector field]]}. See there for more discussion.

\hypertarget{relativistic_form}{}\paragraph*{{Relativistic form}}\label{relativistic_form}

\begin{defn}
\label{PrePlecticForm}\hypertarget{PrePlecticForm}{}
Given a De Donder-Weyl-Hamiltonian $H$, def. \ref{dWHamiltonianByLegendre}, define a [[differential form]]

\begin{displaymath}
\Omega \in \Omega^{n+1}_{cl}((J^1 E)^\ast)
\end{displaymath}
on the dual jet bundle (the \emph{[[multisymplectic form]]} or \emph{[[n-plectic form|pre-(n+1)-plectic form]]}) by

\begin{displaymath}
\Omega 
   \;\coloneqq \;
    \mathbf{d} \phi^i \wedge \mathbf{d} \pi^\mu_i  
   \wedge (\iota_{\partial_\mu} vol_\Sigma) 
    + 
    \mathbf{d} H \wedge vol_\Sigma
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DWHInRelativisticPlecticForm}\hypertarget{DWHInRelativisticPlecticForm}{}
A [[section]] $(\phi^i, \pi^\mu_i) \colon \Sigma \to J^1 E^\ast$ of the dualized first [[jet bundle]] satisfies the [[equations of motion]], prop. \ref{DWHEquations}, precisely if its [[tangent vectors]]

\begin{displaymath}
v_\mu 
    \coloneqq \partial_{\mu} 
  + 
    \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i}
  + 
  \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i}
\end{displaymath}
jointly annihilate the pre-$(n+1)$-plectic form of def. \ref{PrePlecticForm} in that the [[equation]]:

\begin{displaymath}
(\iota_{v_1} \cdots \iota_{v_n}) \Omega
  = 0
\end{displaymath}
holds.

\end{prop}
\begin{proof}
First, the component of this equation which does not contain any $\mathbf{d}\sigma^\mu$ is

\begin{displaymath}
\iota_{v_i}
  \;
  \mathbf{d}\phi^i \wedge \mathbf{d}\pi^i_\mu
  = 
  \mathbf{d}H
  \,.
\end{displaymath}
This is already equivalent to the DWH equation, prop. \ref{DWHEquations}.

Second, this already implies that the components of the equation that are proportional to $\mathbf{d}\sigma^\mu$ are automatically satisfied; because these components are

\begin{displaymath}
\frac{\partial \phi^i}{\partial \sigma^\mu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\nu}
  \mathbf{d}\sigma^\nu
  -
  \frac{\partial \phi^i}{\partial \sigma^\nu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\mu}
  \mathbf{d}\sigma^\nu
  =
  \frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  + 
  \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
\end{displaymath}
and inserting the DWH equations on the left makes the left side identically equal to the right hand side.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Proposition \ref{DWHInRelativisticPlecticForm} is indeed true in the general case where $H$ may be [[spacetime]] dependent (depend nontrivially on the $\sigma^\mu$).

\end{remark}
\hypertarget{nonrelativistic_form}{}\paragraph*{{Non-relativistic form}}\label{nonrelativistic_form}

To obtain the ``non-relativistic'' form of the $(n+1)$-plectic form of the DWH equation, consider the \emph{affine} dual first jet bundle with canonical coordinates $\{\sigma^a , \phi^i, \pi^a_i, e \}$.

\begin{remark}
\label{}\hypertarget{}{}
The canonical [[n-plectic form|pre-(n+1)-plectic form]] on the affine dual first jet bundle is

\begin{displaymath}
\omega 
   \coloneqq 
  \mathbf{d}\phi^i \wedge \mathbf{d}\phi^\mu_a
  \iota_{\partial_{\sigma^\mu}} vol_\Sigma
  + 
  \mathbf{d}e \wedge vol_\Sigma 
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Vector fields tangent to a [[section]] of the affine dual first jet bundle are of the form

\begin{displaymath}
v_\mu 
  \coloneqq 
    \partial_{\mu} 
  + 
    \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i}
  + 
    \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i}
  +
    \frac{\partial e}{\partial \sigma^\mu} \partial_e
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{DWHInNonRelativisticPlecticForm}\hypertarget{DWHInNonRelativisticPlecticForm}{}
On the affine dual first jet bundle the de Donder-Weyl-Hamilton equation characterizes those [[sections]] whose [[tangent vectors]] as above satisfy

\begin{displaymath}
(\iota_{v_1} \cdots \iota_{v_n})
  \omega
  = 
  \mathbf{d}(H + e)
\end{displaymath}
\end{prop}
This has been pointed out in (\hyperlink{Helein02}{H\'e{}lein 02, around equation (4)}).

\begin{proof}
This is a slight variant of the proof of prop. \ref{DWHInRelativisticPlecticForm}. First, the component of the equation independend of $\mathbf{d}\sigma^\nu$ is

\begin{displaymath}
\iota_{v_i}
  \;
  \mathbf{d}\pi^i_\mu \wedge \mathbf{d}\phi^i 
  +
  \mathbf{d}e
  = 
  \mathbf{d}H
  +
  \mathbf{d}e
  \,.
\end{displaymath}
Here the term $\mathbf{d}e$ cancels on both sides and leaves the equation equivalent to the DWH equation as in the first step of the proof of prop. \ref{DWHInRelativisticPlecticForm}.

Second, the component of the claimed equation proportional to $\mathbf{d}\sigma^\mu$ is now

\begin{displaymath}
\frac{\partial \phi^i}{\partial \sigma^\mu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\nu}
  \mathbf{d}\sigma^\nu
  -
  \frac{\partial \phi^i}{\partial \sigma^\nu}
  \frac{\partial \pi_i^\mu}{\partial \sigma^\mu}
  \mathbf{d}\sigma^\nu
  + 
  \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  = 0
  \,.
\end{displaymath}
Inserting into this the DWH equation makes it equivalent to

\begin{displaymath}
\frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  + 
  \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  +   
  \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu
  = 0
\end{displaymath}
This has a unique solution, up to a global constant, given by

\begin{displaymath}
e(\{ \sigma^\nu \}) 
   = 
  - H(
    \{
      \phi^i(\{ \sigma^\nu \}), 
      \pi^\mu_i (\{ \sigma^\nu \})
    \}
    )
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
By the proof of prop. \ref{DWHInNonRelativisticPlecticForm} we have \emph{on shell} that $H + e = 0$ and that $\omega = \Omega$.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hamiltonian n-vector field]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Maybe the first example of what is now called De Donder-Weyl theory appeared in

\begin{itemize}%
\item [[Constantin Carathéodory]], \emph{\"U{}ber die Extremalen und geod\"a{}tischen Felder in der Variationsrechnung der mehrfachen Integrale}, Acta Sci. Math. (Szeged) 4 (1929) 193-216

\end{itemize}
Then Weyl and de Donder independently published

\begin{itemize}%
\item [[Hermann Weyl]], \emph{Geodesic fields in the calculus of variations}, Ann. Math. (2) 36 (1935) 607-629.


\item [[Théophile De Donder]], \emph{Th\'e{}orie invariante du calcul des variations}, Nuov. \'e{}d, Gauthiers--Villars, Paris 1935



\end{itemize}
Reviews include

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/De_Donder%E2%80%93Weyl_theory}{De Donder-Weyl theory}}


\item Narciso Rom\'a{}n-Roy, \emph{Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories}, SIGMA 5 (2009), 100 (\href{http://www.emis.de/journals/SIGMA/2009/100/}{journal}, \href{http://arxiv.org/abs/math-ph/0506022}{arXiv:math-ph/0506022})


\item [[Frédéric Hélein]], \emph{Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory}, (\href{http://arxiv.org/abs/math-ph/0212036}{arXiv:math-ph/0212036})



\end{itemize}
See also

\begin{itemize}%
\item I.V. Kanatchikov, \emph{De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory} , (\href{http://arxiv.org/abs/hep-th/9810165}{arXiv:hep-th/9810165})

\end{itemize}
For more see the references at \emph{[[multisymplectic geometry]]}, at \emph{[[n-plectic geometry]]} and at \emph{[[Hamiltonian n-vector field]]}.

[[!redirects de Donder-Weyl-Hamilton equation]]

[[!redirects De Donder-Weyl hamiltonian]] [[!redirects de Donder-Weyl Hamiltonian]]

[[!redirects Hamilton-De Donder-Weyl equation]] [[!redirects Hamilton-de Donder-Weyl equations]] [[!redirects Hamiltoni-De Donder-Weyl equations]]

[[!redirects De Donder-Weyl equation]] [[!redirects de Donder-Weyl equations]]

[[!redirects De Donder-Weyl formalism]]

[[!redirects Hamilton-De Donder-Weyl equations of motion]] [[!redirects Hamilton-De Donder-Weyl equations of motion]]

[[!redirects de Donder-Weyl theory]]

[[!redirects De Donder-Weyl field theory]]



\end{document}