synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
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Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
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Types of quantum field thories
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, spaces of sections and hence spaces of histories of fields in field theory.
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 of functional 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.
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 the vertical derivative in the variational bicomplex of differential forms on the given jet bundle.
(…)
Exposition of variational calculus in terms of jet bundles and Lepage forms and aimed at examples from physics is in
Fundamental texts on 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. 259€284. (pdf)
Zuckerman’s ideas are used in
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