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\begin{document}

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\section*{evolutionary derivative}

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

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

[[!include variational calculus - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{evolutionary derivative} or ``[[Fréchet derivative]] of a [[tuple]] of differential functions'' (\hyperlink{Olver93}{Olver 93, def. 5.24})) is the [[derivative]] of a [[section]] of some [[vector bundle]] $V$ depending on [[jets]] of a ``[[field bundle]]'' $E$ (def. \ref{FieldDependentSections} below) along the prolongation of an [[evolutionary vector field]] on $E$. Equivalently this is a jet-dependent [[differential operator]] on the [[vertical tangent bundle]] of $E$ and as such is usefully related to the [[Euler-Lagrange derivative]] on $E$ (example \ref{DifferentialOperatorDerivativeOfLagrangianFunction} and prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives} below).

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

In the following \emph{[[fiber bundles]]} are considered in [[differential geometry]] and in particular \emph{[[vector bundle]]} means \emph{[[smooth vector bundle]]}.

\begin{defn}
\label{FieldDependentSections}\hypertarget{FieldDependentSections}{}
\textbf{([[field (physics)|field]]-dependent [[sections]])}

For

\begin{displaymath}
E \overset{fb}{\longrightarrow} \Sigma
\end{displaymath}
a [[fiber bundle]], regarded as a [[field bundle]], and for

\begin{displaymath}
E' \overset{fb'}{\longrightarrow} \Sigma
\end{displaymath}
any other [[fiber bundle]] over the same base space ([[spacetime]]), we write

\begin{displaymath}
\Gamma_{J^\infty_\Sigma(E)}(E')
  \;\coloneqq\;
  \Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' )
  \;=\;
  Hom_\Sigma(J^\infty_\Sigma(E), E')
  \;\simeq\; 
  DiffOp(E,E')
\end{displaymath}
for the [[space of sections]] of the [[pullback of bundles]] of $E'$ to the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$ along $jb$.

\begin{displaymath}
\Gamma_{J^\infty_\Sigma(E)}(E')
  \;=\;
  \left\{
    \itexarray{
       && E'
       \\
       & {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'} 
       \\
       J^\infty_\Sigma(E)
       &\underset{jb}{\longrightarrow}&
       \Sigma
    }
    \phantom{A}\,\,
  \right\}
  \,.
\end{displaymath}
(Equivalently this is the space of [[differential operators]] from sections of $E$ to sections of $E'$. )

\end{defn}
In (\hyperlink{Olver93}{Olver 93, section 5.1, p. 288}) the field dependent sections of def. \ref{FieldDependentSections}, considered in [[local coordinates]], are referred to as [[tuples]] of \emph{differential functions}.

\begin{example}
\label{EvolutionaryVectorFieldsAsFieldDependentSections}\hypertarget{EvolutionaryVectorFieldsAsFieldDependentSections}{}
\textbf{([[source forms]] and [[evolutionary vector fields]] are field-dependent sections)}

For $E \overset{fb}{\to} \Sigma$ a [[field bundle]], write $T_\Sigma E$ for its [[vertical tangent bundle]] and $T_\Sigma^\ast E$ for its [[dual vector bundle]], the [[vertical cotangent bundle]].

Then the field-dependent sections of these bundles according to def. \ref{FieldDependentSections} are identified as follows:

\begin{itemize}%
\item the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of [[evolutionary vector fields]] $v$ as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$:

\begin{displaymath}
\left(
    \itexarray{
      && T_\Sigma E
      \\
      & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}}
      \\
      J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma
    }
  \right)
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
\end{displaymath}

\item $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of [[source forms]] $E$ as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:

\begin{displaymath}
\left(
    \itexarray{
      && T^\ast_\Sigma E
      \\
      & {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}}
      \\
      J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma
    }
  \right)
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
\end{displaymath}


\end{itemize}
This makes manifest the duality pairing between [[source forms]] and [[evolutionary vector fields]]

\begin{displaymath}
\itexarray{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
    \otimes
    \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
    &\longrightarrow&
    C^\infty(J^\infty_\Sigma(E))
  }
\end{displaymath}
which in local coordinates is given by

\begin{displaymath}
(v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a)
  \mapsto
  v^a \omega_a
\end{displaymath}
for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ [[smooth functions]] on the [[jet bundle]].

\end{example}
\begin{defn}
\label{FieldDependentDifferentialOperatorDerivative}\hypertarget{FieldDependentDifferentialOperatorDerivative}{}
\textbf{([[evolutionary derivative of field-dependent section]])}

Let

\begin{displaymath}
E \overset{fb}{\to} \Sigma
\end{displaymath}
be a [[fiber bundle]] regarded as a [[field bundle]] and let

\begin{displaymath}
V \overset{vb}{\to} \Sigma
\end{displaymath}
be a [[vector bundle]]. Then for

\begin{displaymath}
P \in \Gamma_{J^\infty_\Sigma(E)}(V)
\end{displaymath}
a field-dependent section of $E$ accoring to def. \ref{FieldDependentSections}, its \emph{evolutionary derivative} is the morphism

\begin{displaymath}
\itexarray{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
      & \overset{  \mathrm{D}P }{\longrightarrow} &
    \Gamma_{J^\infty_\Sigma(E)}(V)
    \\
    v &\mapsto& \hat v(P) 
  }
\end{displaymath}
which, under the identification of example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}, sense an [[evolutionary vector field]] $v$ to the [[derivative]] of $P$ along the prolongation [[tangent vector field]] $\hat v$ of $v$.

In the case that $E$ and $V$ are [[trivial vector bundles]] over [[Minkowski spacetime]] with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu),  (\rho^b))$, respectively, then this is given by

\begin{displaymath}
((\mathrm{D}P)(v))^b
   \;=\;
  \left(
     v^a \frac{\partial P^b}{\partial \phi^a}
     +
     \frac{d v^a}{d x^\mu}  
     \frac{\partial P^b}{\partial \phi^a_{,\mu}}
     +
      \frac{d^2 v^a}{d x^\mu d x^\nu}
      \frac{\partial  P^b}{\partial \phi^a_{,\mu \nu}}
      +
     \cdots
  \right)
\end{displaymath}
This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent [[differential operator]] from the [[vertical tangent bundle]] $T_\Sigma E$ to $V$, namely a morphism of the form

\begin{displaymath}
\mathrm{D}_P
  \;\colon\;
  J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E
  \longrightarrow
  V
\end{displaymath}
in that

\begin{equation}
\mathrm{D}_P(-,v)
  =
  \mathrm{D}P(v)
  = 
  \hat v (P)
  \,.
\label{FrechetDerivativeAsDifferentialOperatorEquality}\end{equation}
\end{defn}
(\hyperlink{Olver93}{Olver 93, def. 5.24})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{DifferentialOperatorDerivativeOfLagrangianFunction}\hypertarget{DifferentialOperatorDerivativeOfLagrangianFunction}{}
\textbf{([[evolutionary derivative]] of [[Lagrangian function]])}

Over a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] $\Sigma$, let $\mathbf{L} = L dvol \in \Omega^{p,0}_\Sigma(E)$ be a [[Lagrangian density]], with coefficient function regarded as a field-dependent section (def. \ref{FieldDependentSections}) of the [[trivial bundle|trivial]] [[real line bundle]]:

\begin{displaymath}
L \;\in \; \Gamma_{J^\infty_\Sigma}(E)(\Sigma \times \mathbb{R})
  \,,
\end{displaymath}
Then the [[formally adjoint differential operator]]

\begin{displaymath}
(\mathrm{D}_L)^\ast
    \;\colon\;
  J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast
  \longrightarrow
  T_\Sigma^\ast E
\end{displaymath}
of its [[evolutionary derivative]], def. \ref{FieldDependentDifferentialOperatorDerivative}, regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section

\begin{displaymath}
1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)
\end{displaymath}
is the [[Euler-Lagrange derivative]]

\begin{displaymath}
\delta_{EL}\mathbf{L}
  \;=\;
  \left(\mathrm{D}_{L}\right)^\ast(1)
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast)
   \simeq
   \Omega^{p+1,1}_\Sigma(E)_{source}
\end{displaymath}
via the identification from example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}.

\end{example}
(\hyperlink{Olver93}{Olver 93, above (5.80)})

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

\begin{prop}
\label{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}\hypertarget{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}{}
\textbf{([[Euler-Lagrange derivative]] is [[derivation]] via [[evolutionary derivatives]])}

Let $V \overset{vb}{\to} \Sigma$ be a [[vector bundle]] and write $V^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]].

For field-dependent sections (def. \ref{FieldDependentSections})

\begin{displaymath}
\alpha \in \Gamma_{J^\infty_\Sigma(E)}(V)
\end{displaymath}
and

\begin{displaymath}
\beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast)
\end{displaymath}
we have that the [[Euler-Lagrange derivative]] of their canonical pairing to a [[smooth function]] on the [[jet bundle]] is the sum of the derivative of either one via the [[formally adjoint differential operator]] of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) of the other:

\begin{displaymath}
\delta_{EL}( \alpha \cdot \beta^\ast )
  \;=\;
  (\mathrm{D}_\alpha)^\ast(\beta^\ast)
  + 
  (\mathrm{D}_{\beta^\ast})^\ast(\alpha)
\end{displaymath}
\end{prop}
(\hyperlink{Olver93}{Olver 93 (5.80)})

\begin{proof}
It is sufficient to check this in [[local coordinates]]. By the [[product law]] for [[differentiation]] we have

\begin{displaymath}
\begin{aligned}
    \frac{
      \delta_{EL} \left(\alpha \cdot \beta^\ast \right)
    }
    {
      \delta \phi^a
    }
    & =
    \frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a}
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}}
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}}
    \right)  
    -
    \cdots
    \\
    & =
    \phantom{+}
    \frac{\partial \alpha }{\partial \phi^a}
    \cdot \beta^\ast
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \alpha }{\partial \phi^a_{,\mu}}
       \cdot 
       \beta^\ast
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \alpha  }{\partial \phi^a_{,\mu \nu}}
       \cdot 
       \beta^\ast
    \right)  
    -
    \cdots
    \\
    & \phantom{=}
    +
    \frac{\partial \beta^\ast }{\partial \phi^a}
    \cdot \alpha
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}}
       \cdot 
       \alpha
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \beta^\ast  }{\partial \phi^a_{,\mu \nu}}
       \cdot 
       \alpha
    \right)  
    -
    \cdots
    \\
    & =
    (\mathrm{D}_\alpha)^\ast(\beta^\ast)
    + 
    (\mathrm{D}_{\beta^\ast})^\ast(\alpha)
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}\hypertarget{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}{}
\textbf{([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])}

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] over [[Minkowski spacetime]] and regard the [[Euler-Lagrange derivative]]

\begin{displaymath}
\delta_{EL}\mathbf{L}
  \;=\;
  \delta_{EL}L \wedge dvol_\Sigma
\end{displaymath}
as a field-dependent section of the [[vertical cotangent bundle]]

\begin{displaymath}
\delta_{EL}L
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
\end{displaymath}
as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative}) is [[formally self-adjoint differential operator|formally self-adjoint]]:

\begin{displaymath}
(D_{\delta_{EL}L})^\ast
  \;=\;
  D_{\delta_{EL}L}
\end{displaymath}
\end{prop}
(\hyperlink{Olver93}{Olver 93, theorem 5.92})

\begin{prop}
\label{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}\hypertarget{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}{}
\textbf{([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])}

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] over [[Minkowski spacetime]] and regard the [[Euler-Lagrange derivative]]

\begin{displaymath}
\delta_{EL}\mathbf{L}
  \;=\;
  \delta_{EL}L \wedge dvol_\Sigma
\end{displaymath}
as a field-dependent section of the [[vertical cotangent bundle]]

\begin{displaymath}
\delta_{EL}L
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
\end{displaymath}
as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative}) is [[formally self-adjoint differential operator|formally self-adjoint]]:

\begin{displaymath}
(D_{\delta_{EL}L})^\ast
  \;=\;
  D_{\delta_{EL}L}
\end{displaymath}
\end{prop}
(\href{evolutionary+derivative#Olver93}{Olver 93, theorem 5.92}) The following proof is due to [[Igor Khavkine]].

\begin{proof}
By definition of the [[Euler-Lagrange form]] we have

\begin{displaymath}
\frac{\delta_{EL} L }{\delta \phi^a}
  \delta \phi^a
  \, 
  \wedge 
  dvol_\Sigma
  \;=\;
  \delta L \,\wedge dvol_\Sigma \;+\; d(...)
  \,.
\end{displaymath}
Applying the [[variational derivative]] $\delta$ to both sides of this equation yields

\begin{displaymath}
\left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right)
  \wedge
  \delta \phi^a
  \, 
  \wedge 
  dvol_\Sigma
  \;=\;
  \underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma
  \;+\;
  d(...)
  \,.
\end{displaymath}
It follows that for $v,w$ any two [[evolutionary vector fields]] the contraction of their prolongations $\hat v$ and $\hat w$ into the [[variational differential form|differential 2-form]] on the left is

\begin{displaymath}
\left(
    \delta \frac{\delta_{EL} L }{\delta \phi^a}
    \wedge
    \delta \phi^a
  \right)(v,w)
  =
    w^a
    (\mathrm{D}_{\delta_{EL}})_a(v)
    -
    v^b(\mathrm{D}_{\delta_{EL}})_b(w) 
  \,,
\end{displaymath}
by inspection of the definition of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) and their contraction into the form on the right is

\begin{displaymath}
\iota_{\hat v}
  \iota_{\hat w}
  d(...)
  \;=\;
  d(...)
\end{displaymath}
by the fact (prop. \ref{EvolutionaryVectorFieldProlongation}) that contraction with prolongations of evolutionary vector fields coommutes with the [[total spacetime derivative]].

Hence the last two equations combined give

\begin{displaymath}
w^a
  (\mathrm{D}_{\delta_{EL}})_a(v)
  -
  v^b(\mathrm{D}_{\delta_{EL}})_b(w) 
  \;=\;
  d(...)
  \,.
\end{displaymath}
This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be [[formally self-adjoint differential operator]].

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

\begin{itemize}%
\item [[Peter Olver]], \emph{Applications of Lie groups to Differential equations}, Graduate Texts in Mathematics, Springer 1993


\item [[Glenn Barnich]], equation(3) of \emph{A note on gauge systems from the point of view of Lie algebroids}, in P. Kielanowski, V. Buchstaber, A. Odzijewicz, M.. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (\href{https://arxiv.org/abs/1010.0899}{arXiv:1010.0899}, \href{https://doi.org/10.1063/1.3527427}{doi:/10.1063/1.3527427})


\item [[Igor Khavkine]], starting with p. 45 of \emph{Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory} (\href{https://arxiv.org/abs/1211.1914}{arXiv:1211.1914})



\end{itemize}
[[!redirects evolutionary derivatives]]

[[!redirects evolutionary derivative of field-dependent section]] [[!redirects evoolutionary derivatives of field-dependent sections]]



\end{document}