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\section*{variational calculus}

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

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\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{InTermsOfSmoothSpaces}{In terms of smooth spaces}\dotfill \pageref*{InTermsOfSmoothSpaces} \linebreak
\noindent\hyperlink{SmoothFunctionals}{Smooth functionals}\dotfill \pageref*{SmoothFunctionals} \linebreak
\noindent\hyperlink{FunctionalDerivative}{Functional derivative}\dotfill \pageref*{FunctionalDerivative} \linebreak
\noindent\hyperlink{in_terms_of_the_variational_bicomplex}{In terms of the variational bicomplex}\dotfill \pageref*{in_terms_of_the_variational_bicomplex} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{relation_to_covariant_phase_spaces}{Relation to covariant phase spaces}\dotfill \pageref*{relation_to_covariant_phase_spaces} \linebreak
\noindent\hyperlink{by_functorial_analysis_and_geometry}{By functorial analysis and $\mathcal{D}$-geometry}\dotfill \pageref*{by_functorial_analysis_and_geometry} \linebreak


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

\textbf{Variational calculus} -- sometimes called \textbf{secondary calculus} -- is a version of [[differential calculus]] that deals with local extremization of [[nonlinear functionals]]: extremization of differentiable functions on non-finite dimensional spaces such as [[mapping spaces]], [[spaces of sections]] and hence [[spaces of histories]] of [[field (physics)|fields]] in [[field theory]].

Specifically, it studies the \emph{[[critical points]]} , i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of [[section]]s of [[jet bundle]]s. The kinds of equations specifying these critical points are [[Euler-Lagrange equation]]s.

This applies to, and is largely motivated from, the study of [[action functional]]s in [[physics]]. In [[classical physics]] the critical points of a specified action functional on the space of field configurations encode the physically observable configurations.

There are strong [[cohomology|cohomological]] tools for studying variational calculus, such as the [[variational bicomplex]] and [[BV-BRST formalism]].

\hypertarget{InTermsOfSmoothSpaces}{}\subsection*{{In terms of smooth spaces}}\label{InTermsOfSmoothSpaces}

We discuss some basics of variational calculus of functional in terms of [[smooth spaces]] and in particular in terms of [[diffeological spaces]].

\hypertarget{SmoothFunctionals}{}\subsubsection*{{Smooth functionals}}\label{SmoothFunctionals}

Let $X$ be a [[smooth manifold]]. Let $\Sigma$ be a [[smooth manifold|smooth]] [[manifold with boundary]] $\partial \Sigma \hookrightarrow \Sigma$.

Write

\begin{displaymath}
[\Sigma, X] \in Smooth0Type
\end{displaymath}
for the [[smooth space]] (a [[diffeological space]]) which is the [[mapping space]] from $\Sigma$ to $X$, hence such that for each $U \in$ [[CartSp]] we have

\begin{displaymath}
[\Sigma, X](U) = C^\infty(U \times \Sigma, X)
  \,.
\end{displaymath}
\begin{defn}
\label{MappingSpaceWithNonVaryingBoundary}\hypertarget{MappingSpaceWithNonVaryingBoundary}{}
Write

\begin{displaymath}
[\Sigma, X]_{\partial \Sigma}
  \coloneqq
  [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]
\end{displaymath}
for the [[pullback]] in smooth spaces

\begin{displaymath}
\itexarray{
    [\Sigma,X]_{\partial \Sigma}
    &\to&
    \flat [\partial \Sigma, X]
    \\
    \downarrow && \downarrow
    \\
    [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X]
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;


\item the right vertical morphism is the [[counit of an adjunction|counit]] of the $(Disc \dashv \Gamma)$-[[adjunction]] on smooth spaces.



\end{itemize}
\end{defn}
\begin{prop}
\label{PlotsOfMappingSpaceWithNonVaryingBoundary}\hypertarget{PlotsOfMappingSpaceWithNonVaryingBoundary}{}
The [[smooth space]] $[\Sigma, X]_{\partial \Sigma}$ is a [[diffeological space]] whose underlying set is $C^\infty(\Sigma,X)$ and whose $U$-plots for $U \in$ [[CartSp]] are smooth functions $\phi \colon U \times \Sigma \to X$ such that $\phi(-,s) \colon U \to X$ is the constant function for all $s \in \partial \Sigma \hookrightarrow \Sigma$.

\end{prop}
\begin{defn}
\label{SmoothFunctional}\hypertarget{SmoothFunctional}{}
A \textbf{functional} on the mapping space $[\Sigma, X]$ is a [[homomorphism]] of smooth spaces

\begin{displaymath}
S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{FunctionalDerivative}{}\subsubsection*{{Functional derivative}}\label{FunctionalDerivative}

Write

\begin{displaymath}
\mathbf{d} \colon \mathbb{R} \to \Omega^1
\end{displaymath}
for the [[de Rham differential]] incarnated as a [[homomorphism]] of [[smooth spaces]] from the [[real line]] to the smooth [[moduli space]] of [[differential 1-forms]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[functional derivative]]}

\begin{displaymath}
\mathbf{d}S
  \in 
  \Omega^1([\Sigma,X]_{\partial \Sigma})
\end{displaymath}
of a functional $S$, def. \ref{SmoothFunctional}, is simply its [[de Rham differential]] as a homomorphism of smooth spaces, hence the composite

\begin{displaymath}
\mathbf{d} S \colon [
  \Sigma, X]_{\partial \Sigma} 
     \stackrel{S}{\to} 
  \mathbb{R}   
    \stackrel{\mathbf{d}}{\to}
  \Omega^1
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
This means that for each smooth parameter space $U \in$ [[CartSp]] and for each smooth $U$-parameterized collection

\begin{displaymath}
\phi \colon U \times \Sigma \to X
\end{displaymath}
of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

\begin{displaymath}
S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})
\end{displaymath}
is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

\begin{displaymath}
(\mathbf{d}S)_\phi \in \Omega^1(U)
\end{displaymath}
is the smooth [[differential 1-form]]

\begin{displaymath}
(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi))
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard [[interval]]. Let

\begin{displaymath}
L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))
\end{displaymath}
and consider the functional

\begin{displaymath}
S
  \colon
  ([0,1] \stackrel{\gamma}{\to} X)
  \mapsto
  \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t
  \,.
\end{displaymath}
Then for $U = \mathbb{R}$ and any smooth $U$-parameterized collection $\{\gamma_{u} \colon \Sigma \to X\}_{u \in I}$ the functional derivative takes the value

\begin{displaymath}
\begin{aligned}
   \mathbf{d}S_{\gamma_{(-)}}
   & =
   \left(
     \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt
   \right)
   \mathbf{d}u
   \\
    & = 
     \int_{0}^1 
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      + 
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial \dot \gamma_u(t)}{\partial u}
     \right)
     \mathbf{d} u
     \\
     & =
     \int_{0}^1 
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      + 
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u}
     \right)
     \mathbf{d} u
     \\
     & =
     \int_{0}^1
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
      - 
       \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
     \right)
     \frac{\partial \gamma_u(s)}{\partial u}
     \mathbf{d}u
  \end{aligned}
  \,.
\end{displaymath}
Here we used [[integration by parts]] and used that the boundary term vanishes

\begin{displaymath}
\begin{aligned}
    \int_{0}^1 \frac{\partial}{\partial t} 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s))
      \frac{\partial \gamma_u(s)}{\partial u}
    \right)
    d s
    & = 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1))
      \frac{\partial \gamma_u(1)}{\partial u}
    \right)
    - 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0))
      \frac{\partial \gamma_u(0)}{\partial u}
    \right)
    \\
    & = 0
  \end{aligned}
\end{displaymath}
since by prop. \ref{PlotsOfMappingSpaceWithNonVaryingBoundary} $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.

\end{example}
\hypertarget{in_terms_of_the_variational_bicomplex}{}\subsection*{{In terms of the variational bicomplex}}\label{in_terms_of_the_variational_bicomplex}

In the special case that the functional to be varied comes from a [[Lagrangian density]], then its variational derivative is the image under [[transgression of variational differential forms|transgression]] of the [[vertical derivative]] in the [[variational bicomplex]] of differential forms on the given [[jet bundle]].

(\ldots{})

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

\begin{itemize}%
\item [[Euler-Lagrange equations]], [[shell]]


\item [[source form]], [[evolutionary vector field]], [[evolutionary derivative]]


\item [[de Donder-Weyl formalism]]


\item [[phase space]]


\item [[variational bicomplex]], [[secondary differential calculus]]


\item [[Lagrange multiplier]]


\item [[variational sequence]]



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

Exposition of variational calculus in terms of [[jet bundles]] and [[Lepage forms]] and aimed at examples from [[physics]] is in

\begin{itemize}%
\item [[Jana Musilová]], [[Stanislav Hronek]], \emph{The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories}, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (\href{https://doi.org/10.1515/cm-2016-0012}{doi.org/10.1515/cm-2016-0012})

\end{itemize}
Fundamental texts on variational calculus include

\begin{itemize}%
\item [[Ian Anderson]], \emph{The variational bicomplex}, ([[AndersonVariationalBicomplex.pdf:file]])


\item Hubert Goldschmidt, [[Shlomo Sternberg]], \emph{The Hamilton-Cartan formalism in the calculus of variations}, Annales de l'institut Fourier 23 no. 1 (1973), p. 203-267 \href{http://www.numdam.org/item?id=AIF_1973__23_1_203_0}{numdam}


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


\item [[Demeter Krupka]], \emph{Introduction to global variational geometry}, 2015


\item Olga Krupkov\'a{}, \emph{The geometry of ordinary variational equations}, Springer 1997, 251 p.


\item Robert Hermann, \emph{Some differential-geometric aspects of the Lagrange variational problem}, Illinois J. Math. \textbf{6}, 1962, 634--673 \href{http://www.ams.org/mathscinet-getitem?mr=145457}{MR145457} \href{http://projecteuclid.org/euclid.ijm/1255632711}{euclid}; \emph{Differential geometry and the calculus of variations}, Acad. Press 1968


\item J. Jost, X. Li-Jost, \emph{Calculus of variations}, CUP 1998


\item [[Gregg Zuckerman|G. J. Zuckerman]], \emph{Action Principles and Global Geometry} , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259\euro{}284. ([[ZuckermanVariation.pdf:file]])



\end{itemize}
Zuckerman's ideas are used in

\begin{itemize}%
\item [[Marco Benini]], [[Alexander Schenkel]], \emph{Poisson algebras for non-linear field theories in the Cahiers topos}, \href{http://arxiv.org/abs/1602.00708}{arxiv/1602.00708}

\end{itemize}
Examples: [[Jürgen Jost]], \emph{Variational problems from physics and geometry}, \href{http://www.mis.mpg.de/fileadmin/jjost/variational_problems_from_physics_and_geometry.pdf}{pdf}

\begin{itemize}%
\item J. J. Duistermaat, \emph{On the Morse index in variational calculus}, Adv. Math. \textbf{21} (1976), 2, 173--195 \href{http://www.maths.ed.ac.uk/~aar/papers/duistermaat.pdf}{pdf}.

\end{itemize}
Some new results are in

\begin{itemize}%
\item [[E. Getzler]], \emph{A Darboux theorem for Hamiltonian operators in the formal calculus of variations}, Duke Math. J. \textbf{111}, n. 3 (2002), 535-560, \href{http://www.ams.org/mathscinet-getitem?mr=1885831}{MR2003e:32026}, \href{http://dx.doi.org/10.1215/S0012-7094-02-11136-3}{doi}
\item Alberto De Sole, Victor G. Kac, \emph{The variational Poisson cohomology}, \href{http://arxiv.org/abs/1106.0082}{arxiv/1106.0082}

\end{itemize}
Geometric extremization problems (e.g. minimal surfaces), see also [[geometric measure theory]]:

\begin{itemize}%
\item [[Jürgen Jost]], \emph{The geometric calculus of variations: a short survey and a list of open problems}, Exposition. Math. \textbf{6} (1988), no. 2, 111--143, \href{http://www.ams.org/mathscinet-getitem?mr=938179}{MR89h:58036}
\item H. Federer, \emph{Geometric measure theory}, Springer 1969(especially appendices to Russian transl.)
\item Frederick J., Jr. Almgren, Almgren's big regularity paper (book form of a 1970s preprint)

\end{itemize}
Discussion in the context of BV formalism:

\begin{itemize}%
\item Arthemy V. Kiselev, \emph{The geometry of variations in Batalin-Vilkovisky formalism}, \href{http://arxiv.org/abs/1312.1262}{http://arxiv.org/abs/1312.1262}

\end{itemize}
Other references

\begin{itemize}%
\item J. C. P. Bus, \emph{The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems}, SIAM J. Control and Optimization \textbf{22}, 5, 1984, 740-757 \href{http://oai.cwi.nl/oai/asset/2552/2552A.pdf}{pdf}

\end{itemize}
\hypertarget{relation_to_covariant_phase_spaces}{}\paragraph*{{Relation to covariant phase spaces}}\label{relation_to_covariant_phase_spaces}

\begin{itemize}%
\item L. Vitagliano, \emph{Secondary calculus and the covariant phase space}, \href{http://diffiety.ac.ru/preprint/2008/02-08.pdf}{pdf}

\end{itemize}
\hypertarget{by_functorial_analysis_and_geometry}{}\subsubsection*{{By functorial analysis and $\mathcal{D}$-geometry}}\label{by_functorial_analysis_and_geometry}

See also references at [[diffiety]].

A formalism for variational calculus based on [[functorial analysis]] (with a precise relation with functional analytic methods and jet formalism) and a long list of examples of variational problems arising in [[classical mechanics]] and [[quantum field theory]] are collected in

\begin{itemize}%
\item [[Frédéric Paugam]], \emph{Towards the mathematics of quantum field theory} (\href{http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf}{pdf})

\end{itemize}
The formulation of variational calculus in terms of [[diffeological space]]s is mentioned for instance in section 1.65 of

\begin{itemize}%
\item [[Patrick Iglesias-Zemmour]], \emph{Diffeology} (\href{http://math.huji.ac.il/~piz/documents/Diffeology.pdf#page=64}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Frédéric Paugam]], \emph{Histories and observables in covariant field theory} (\href{http://arxiv.org/abs/1010.3210}{arXiv:1010.3210}), sec. 2.4

\end{itemize}
following section 2.3.20 of

\begin{itemize}%
\item [[Alexander Beilinson]], [[Vladimir Drinfeld]], \emph{[[Chiral Algebras]]}

\end{itemize}
For variational calculus in [[nonstandard analysis]] see survey

\begin{itemize}%
\item V\'i{}tor Neves, \emph{Nonstandard calculus of variations, a survey}, \href{http://www2.mat.ua.pt/vneves/nsa/CalcVar-vitor6.pdf}{pdf}

\end{itemize}
category: analysis, physics [[!redirects calculus of variations]] [[!redirects variational derivative]] [[!redirects variational derivatives]]



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