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Variational calculus – sometimes called 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.
Specifically, it studies the critical points , i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of sections of jet bundles. The kinds of equations specifying these critical points are Euler-Lagrange equations.
This applies to, and is largely motivated from, the study of action functionals 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 cohomological tools for studying variational calculus, such as the variational bicomplex and BV-BRST formalism.
We discuss some basics of variational calculus in terms of smooth spaces and in particular in terms of diffeological spaces.
Let $X$ be a smooth manifold. Let $\Sigma$ be a smooth manifold with boundary $\partial \Sigma \hookrightarrow \Sigma$.
Write
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
Write
for the pullback in smooth spaces
where
the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;
the right vertical morphism is the counit of the $(Disc \dashv \Gamma)$-adjunction on smooth spaces.
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 $s \in \partial \Sigma \hookrightarrow \Sigma$.
A functional on the mapping space $[\Sigma, X]$ is a homomorphism of smooth spaces
Write
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.
of a functional $S$, def. 2, is simply its de Rham differential as a homomorphism of smooth spaces, hence the composite
This means that for each smooth parameter space $U \in$ CartSp and for each smooth $U$-parameterized collection
of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,
is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then
is the smooth differential 1-form
Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard interval. Let
and consider the functional
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
Here we used integration by parts? and used that the boundary term vanishes
since by prop. 1 $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.
Fundamental texts of variational calculus include
Ian Anderson, The variational bicomplex, pdf
Hubert Goldschmidt, Shlomo Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Annales de l’institut Fourier 23 no. 1 (1973), p. 203-267 numdam
Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.
Demeter Krupka, Introduction to global variational geometry, 2015
Olga Krupková, The geometry of ordinary variational equations, Springer 1997, 251 p.
Robert Hermann, Some differential-geometric aspects of the Lagrange variational problem, Illinois J. Math. 6, 1962, 634–673 MR145457 euclid; Differential geometry and the calculus of variations, Acad. Press 1968
J. Jost, X. Li-Jost, Calculus of variations, CUP 1998
G. J. Zuckerman, Action Principles and Global Geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259284. (pdf)
Examples: Jürgen Jost, Variational problems from physics and geometry, pdf
Some new results are in
Geometric extremization problems (e.g. minimal surfaces), see also geometric measure theory:
Discussion in the context of BV formalism:
Other references
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
The formulation of variational calculus in terms of diffeological spaces is mentioned for instance in section 1.65 of
following section 2.3.20 of
For variational calculus in nonstandard analysis see survey