# nLab variational calculus

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Variational calculus

variational calculus

# Contents

## 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.

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 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 $\partial \Sigma ↪\Sigma$.

Write

$\left[\Sigma ,X\right]\in \mathrm{Smooth}0\mathrm{Type}$[\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

$\left[\Sigma ,X\right]\left(U\right)={C}^{\infty }\left(U×\Sigma ,X\right)\phantom{\rule{thinmathspace}{0ex}}.$[\Sigma, X](U) = C^\infty(U \times \Sigma, X) \,.
###### Definition

Write

$\left[\Sigma ,X{\right]}_{\partial \Sigma }≔\left[\Sigma ,X\right]{×}_{\left[\partial \Sigma ,X\right]}♭\left[\partial \Sigma ,X\right]$[\Sigma, X]_{\partial \Sigma} \coloneqq [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]

for the pullback in smooth spaces

$\begin{array}{ccc}\left[\Sigma ,X{\right]}_{\partial \Sigma }& \to & ♭\left[\partial \Sigma ,X\right]\\ ↓& & ↓\\ \left[\Sigma ,X\right]& \stackrel{\left(-\right){\mid }_{\partial \Sigma }}{\to }& \left[\partial \Sigma ,X\right]\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 $\left[\partial \Sigma ↪\Sigma ,X\right]$ of configurations to the boundary;

• the right vertical morphism is the counit of the $\left(\mathrm{Disc}⊣\Gamma \right)$-adjunction on smooth spaces.

###### Proposition

The smooth space $\left[\Sigma ,X{\right]}_{\partial \Sigma }$ is a diffeological space whose underlying set is ${C}^{\infty }\left(\Sigma ,X\right)$ and whose $U$-plots for $U\in$ CartSp are smooth functions $\varphi :U×\Sigma \to X$ such that $\varphi \left(-,s\right):U\to X$ is the constant function for all $s\in \partial \Sigma ↪\Sigma$.

###### Definition

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

$S:\left[\Sigma ,X{\right]}_{\partial \Sigma }\to ℝ\phantom{\rule{thinmathspace}{0ex}}.$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} \,.

### Functional derivative

Write

$d:ℝ\to {\Omega }^{1}$\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

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

of a functional $S$, def. 2, is simply its de Rham differential as a homomorphism of smooth spaces, hence the composite

$dS:\left[\Sigma ,X{\right]}_{\partial \Sigma }\stackrel{S}{\to }ℝ\stackrel{d}{\to }{\Omega }^{1}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\varphi :U×\Sigma \to X$\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 $\Sigma$,

${S}_{\varphi }:\left[\Sigma ,X{\right]}_{\partial \Sigma }\left(U\right)\to {C}^{\infty }\left(U,ℝ\right)$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

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

is the smooth differential 1-form

$\left(dS{\right)}_{\varphi }=d\left(S\left(\varphi \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi)) \,.
###### Example

Let $\Sigma =\left[0,1\right]↪ℝ$ be the standard interval. Let

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

and consider the functional

$S:\left(\left[0,1\right]\stackrel{\gamma }{\to }X\right)↦{\int }_{0}^{1}L\left(\gamma \left(t\right),\stackrel{˙}{\gamma }\left(t\right)\right)dt\phantom{\rule{thinmathspace}{0ex}}.$S \colon ([0,1] \stackrel{\gamma}{\to} X) \mapsto \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t \,.

Then for $U=ℝ$ and any smooth $U$-parameterized collection $\left\{{\gamma }_{u}:\Sigma \to X{\right\}}_{u\in I}$ the functional derivative takes the value

$\begin{array}{rl}d{S}_{{\gamma }_{\left(-\right)}}& =\left(\frac{d}{du}{\int }_{0}^{1}L\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\mathrm{dt}\right)du\\ & ={\int }_{0}^{1}\left(\frac{\partial L}{\partial \gamma }\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\frac{d{\gamma }_{u}\left(t\right)}{du}+\frac{\partial L}{\partial \stackrel{˙}{\gamma }}\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\frac{\partial {\stackrel{˙}{\gamma }}_{u}\left(t\right)}{\partial u}\right)du\\ & ={\int }_{0}^{1}\left(\frac{\partial L}{\partial \gamma }\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\frac{d{\gamma }_{u}\left(t\right)}{du}+\frac{\partial L}{\partial \stackrel{˙}{\gamma }}\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\frac{\partial }{\partial t}\frac{\partial {\gamma }_{u}\left(t\right)}{\partial u}\right)du\\ & ={\int }_{0}^{1}\left(\frac{\partial L}{\partial \gamma }\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)-\frac{\partial }{\partial t}\frac{\partial L}{\partial \stackrel{˙}{\gamma }}\left({\gamma }_{u}\left(t\right),{\stackrel{˙}{\gamma }}_{u}\left(t\right)\right)\right)\frac{\partial {\gamma }_{u}\left(s\right)}{\partial u}du\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{rl}{\int }_{0}^{1}\frac{\partial }{\partial t}\left(\frac{\partial }{\partial \stackrel{˙}{\gamma }}L\left({\gamma }_{u}\left(s\right),{\stackrel{˙}{\gamma }}_{u}\left(s\right)\right)\frac{\partial {\gamma }_{u}\left(s\right)}{\partial u}\right)ds& =\left(\frac{\partial }{\partial \stackrel{˙}{\gamma }}L\left({\gamma }_{u}\left(1\right),{\stackrel{˙}{\gamma }}_{u}\left(1\right)\right)\frac{\partial {\gamma }_{u}\left(1\right)}{\partial u}\right)-\left(\frac{\partial }{\partial \stackrel{˙}{\gamma }}L\left({\gamma }_{u}\left(0\right),{\stackrel{˙}{\gamma }}_{u}\left(0\right)\right)\frac{\partial {\gamma }_{u}\left(0\right)}{\partial u}\right)\\ & =0\end{array}$\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. 1 ${\gamma }_{\left(-\right)}\left(1\right)$ and ${\gamma }_{\left(-\right)}\left(0\right)$ are constant.

## References

Fundamental texts of variational calculus include

• Ian Anderson, The variational bicomplex, pdf

• Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

• 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

• Robert Hermann, 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)

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

• 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)

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

#### Relation to covariant phase spaces

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