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\begin{document}

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\section*{shell}

\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{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[physics]] the [[critical locus]] of the [[action functional]], hence the [[solution]] space to the [[partial differential equation|partial differential]] [[equations of motion]] is often called the \emph{shell}, or the \emph{physical shell} (the [[diffiety]] corresponding to the PDE of motion).

This terminology derives from the case of the free [[relativistic particle]] propagating in [[Minkowski spacetime]], for which the space of solutions is the hyperbola of vectors whose [[Minkowski metric|Minkowski norm square]] is the [[mass]] of the particle. This hyperbola is naturally called the \emph{mass shell}.

It is common to use the adjective ``on-shell'' for statements that apply after restriction to the shell, and ``off-shell'' for statements that apply generally.

For instance [[Noether's theorem]] gives for each symmetry of a [[local Lagrangian]] an \emph{on-shell [[conserved current]]}, namely a differential form on spacetime, depending on the field configurations, which is closed (only, in general) if the given field configuration satisfies its equations of motion. Such a differential form which is closed even if the equations of motion do not hold would be called an \emph{off-shell conserved current}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

In terms of [[variational calculus]] the field configurations are encoded by a [[field bundle]] $E$ over [[spacetime]]/[[worldvolume]] $\Sigma$ as being the [[sections]] of this bundle. The shell then is encoded by a sub-bundle

\begin{displaymath}
\itexarray{
    \mathcal{E} &&\hookrightarrow && J^\infty_\Sigma E
    \\
    & \searrow && \swarrow
    \\
    && \Sigma
  }
\end{displaymath}
of the [[jet bundle]] $J^\infty_\Sigma E$ of $E$, namely the subbundle of those pointwise field configurations with those jets (derivatives) that do solve the equations of motion (see at \emph{[[diffiety]]}).

This way a field configuration given by a section $\phi$ of $E$ is a solution to the equations of motion precisely if its [[jet prolongation]], being a section of $J^\infty_\Sigma E$ comes from a section of $\mathcal{E}$ under this inclusion.

If one thinks of all this not as happening in the [[category]] of [[bundles]] over $\Sigma$ but in the category $PDE_\Sigma$ of [[partial differential equations]] over $\Sigma$, then the above inclusion becomes simply $\iota \colon \mathcal{E} \hookrightarrow E$, where now $E$ is thought of as the trivial partial differential equation on its sections, the one for which each section is a solution. Then a field configuration is just a morphism $\phi : \Sigma \longrightarrow E$ in $PDE_\Sigma$, and this is a solution precisely if it factors through the inclusion of the shell:

\begin{displaymath}
\itexarray{
    && \mathcal{E}
    \\
    & {}^\mathllap{\phi_{sol}}{\nearrow} & \downarrow^{\mathrlap{\iota}}
    \\
    \Sigma &\underset{\phi}{\longrightarrow}& E
  }
\end{displaymath}
Accordingly, a [[variational bicomplex|horizontal form]] $J \in \Omega^{p}_H(E) \hookrightarrow \Omega^{p+1}(J^\infty_\Sigma E)$ is an \emph{on-shell [[conserved current]]} if it becomes horizontally closed after restriction to the shell:

\begin{displaymath}
\iota^\ast d_H J = 0
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

For $(E,\mathbf{L})$ a [[Lagrangian field theory]], let $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ be the prolonged shell.

The following is intuitively obvious but not entirely trivial to prove:

\begin{prop}
\label{InfinitesialSymmetryOfTheLagrangianIsAlsoSymmetryOfTheShell}\hypertarget{InfinitesialSymmetryOfTheLagrangianIsAlsoSymmetryOfTheShell}{}
\textbf{([[infinitesimal symmetry of the Lagrangian]] is also symmetry of the shell)}

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]]. Then if $\hat v$ is the prolongation of an [[evolutionary vector field]] which is an [[infinitesimal symmetry of the Lagrangian]] in that the [[Lie derivative]] of $\mathbf{L}$ along $\hat v$ is [[horizontal derivative|horizontally exact]], then the [[flow]] of $\hat v$ preserves the prolonged shell.

\end{prop}
(\hyperlink{Olver95}{Olver 95, theorem 5.53})

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

\begin{itemize}%
\item \emph{\href{https://www.physicsforums.com/insights/higher-prequantum-geometry-ii-principle-extremal-action-comonadically/}{Higher prequantum geometry II: The principle of extremal action -- comonadically}}


\item [[Peter Olver]], \emph{Applications of Lie groups to differential equations}, Springer; \emph{Equivalence, invariants, and symmetry}, Cambridge Univ. Press 1995.



\end{itemize}
[[!redirects shells]]

[[!redirects on-shell]] [[!redirects off-shell]]

[[!redirects mass shell]] [[!redirects mass shells]]

[[!redirects prolonged shell]]



\end{document}