nLab variational calculus

Idea

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.

In terms of smooth spaces

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

Smooth functionals

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

Write

$$[\Sigma, X] \in Smooth0Type$$

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

$$[\Sigma, X](U) = C^\infty(U \times \Sigma, X) \,.$$

Definition

Write

$$[\Sigma, X]_{\partial \Sigma} \coloneqq [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]$$

for the pullback in smooth spaces

$$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,$$

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.

Proposition

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 $s \in \partial \Sigma \hookrightarrow \Sigma$.

Definition

A functional on the mapping space $[\Sigma, X]$ is a homomorphism of smooth spaces

$$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} \,.$$

Functional derivative

Write

$$\mathbf{d} \colon \mathbb{R} \to \Omega^1$$

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.

Definition

The functional derivative

$$\mathbf{d}S \in \Omega^1([\Sigma,X]_{\partial \Sigma})$$

of a functional $S$, def. , is simply its de Rham differential as a homomorphism of smooth spaces, hence the composite

$$\mathbf{d} S \colon [ \Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$$

Definition

This means that for each smooth parameter space $U \in$ CartSp and for each smooth $U$-parameterized collection

$$\phi \colon U \times \Sigma \to X$$

of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

$$S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})$$

is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

$$(\mathbf{d}S)_\phi \in \Omega^1(U)$$

is the smooth differential 1-form

$$(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi)) \,.$$

Example

Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard interval. Let

$$L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))$$

and consider the functional

$$S \colon ([0,1] \stackrel{\gamma}{\to} X) \mapsto \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t \,.$$

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

$$\begin{aligned} \mathbf{d}S_{\gamma_{(-)}} & = \left( \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt \right) \mathbf{d}u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial \dot \gamma_u(t)}{\partial u} \right) \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u} \right) \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) - \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \right) \frac{\partial \gamma_u(s)}{\partial u} \mathbf{d}u \end{aligned} \,.$$

Here we used integration by parts and used that the boundary term vanishes

$$\begin{aligned} \int_{0}^1 \frac{\partial}{\partial t} \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s)) \frac{\partial \gamma_u(s)}{\partial u} \right) d s & = \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1)) \frac{\partial \gamma_u(1)}{\partial u} \right) - \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0)) \frac{\partial \gamma_u(0)}{\partial u} \right) \\ & = 0 \end{aligned}$$

since by prop. $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.

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 the vertical derivative in the variational bicomplex of differential forms on the given jet bundle.

(…)

Related concepts

• Euler-Lagrange equations, shell

• source form, evolutionary vector field, evolutionary derivative

• de Donder-Weyl formalism

• phase space

• variational bicomplex, secondary differential calculus

• Lagrange multiplier

• variational sequence

References

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

• Jana Musilová, Stanislav Hronek, 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) (doi.org/10.1515/cm-2016-0012)

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

• Marco Benini, Alexander Schenkel, Poisson algebras for non-linear field theories in the Cahiers topos, arxiv/1602.00708

Examples: Jürgen Jost, Variational problems from physics and geometry, pdf

• J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math. 21 (1976), 2, 173–195 pdf.

Some new results are in

• E. Getzler, A Darboux theorem for Hamiltonian operators in the formal calculus of variations, Duke Math. J. 111, n. 3 (2002), 535-560, MR2003e:32026, doi
• Alberto De Sole, Victor G. Kac, The variational Poisson cohomology, arxiv/1106.0082

Geometric extremization problems (e.g. minimal surfaces), see also geometric measure theory:

• Jürgen Jost, The geometric calculus of variations: a short survey and a list of open problems, Exposition. Math. 6 (1988), no. 2, 111–143, MR89h:58036
• H. Federer, Geometric measure theory, Springer 1969(especially appendices to Russian transl.)
• Frederick J., Jr. Almgren, Almgren’s big regularity paper (book form of a 1970s preprint)

Discussion in the context of BV formalism:

• Arthemy V. Kiselev, The geometry of variations in Batalin-Vilkovisky formalism, http://arxiv.org/abs/1312.1262

Other references

• J. C. P. Bus, The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems, SIAM J. Control and Optimization 22, 5, 1984, 740-757 pdf

In the covariant phase space-perspective:

• L. Vitagliano, Secondary calculus and the covariant phase space, pdf

On smooth sets as a convenient category for variational calculus of Lagrangian classical field theory:

• Grigorios Giotopoulos, Hisham Sati, Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]

By functorial analysis and $\mathcal{D}$-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

• Frédéric Paugam, Towards the mathematics of quantum field theory (pdf)

The formulation of variational calculus in terms of diffeological spaces is mentioned for instance in section 1.65 of

• Patrick Iglesias-Zemmour, Diffeology (pdf)

• Frédéric Paugam, Histories and observables in covariant field theory (arXiv:1010.3210), sec. 2.4

following section 2.3.20 of

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras

For variational calculus in nonstandard analysis see survey

• Vítor Neves, Nonstandard calculus of variations, a survey, pdf

category: analysis, physics