Functional

Prerequisites: Calculus.

• Intuitive
• Concrete
• Abstract

Let $\Omega(\mathcal{Q})$ be the set of functions $q:\mathbb{R^n} \to \mathcal{Q}$, then a functional S is a map

$$S:\Omega \to \mathbb{R}; S[q] \mapsto \alpha \in\mathbb{R} .$$

So we can see how a functional is a function of functions as we said before, this is the reason why the notation $S[\cdot]$ instead of $S(\cdot)$, to remind that it is more that the eyes meet.

A functional might also be dependent on the derivatives of a function, to an arbitrary order.

What interest us

In variational calculus, for $n=1$ and $q$ defined on an interval $[a,b]$, the usual form that a functional takes is that of an integral of an algebraic combination of the a function and its derivatives:

$$S[q] = \int_a^b L(q(x), q'(x), q''(x), \dots, x)d x$$

with $L: T^{(n)}\mathcal Q \times [a,b] \to \mathbb R$. Note that $L$ is a function from a manifold to the reals. And what is integrated is $L\circ \Gamma q(x)$, where $\Gamma$ is the lift of the function $q$ to its fibres.

But there can be other functionals: maximum/minimum value of a function, evaluation of the function at a point (i.e. $q \mapsto q(x_0)$ is also a functional)…

Intuitive

An ordinary function takes a number and returns a number: put in a time, get out a position. A functional takes a whole function and returns a single number. Put in the complete shape of a curve, get out its length. Put in the complete path of a ball between two moments, get out the time it took, or the energy it used.

A good picture is a judge who scores entire performances. The judge does not care what happened at one particular instant; the score depends on everything that happened from beginning to end. Change the performance a little anywhere and the score may change.

Functionals are the natural objects whenever we ask which curve, shape or field is best in some sense. The shortest path between two points is the curve that makes the length functional smallest. In classical mechanics the action is a functional of the path, and the path the system actually follows is the one where the action does not change under small deformations. Finding such special functions is the subject of variational calculus.

Concrete

Examples

• Length of the graph of $y(x)$ between $x = a$ and $x = b$: $\displaystyle L[y] = \int_a^b \sqrt{1 + y'(x)^2}\, dx$.
• Action of a particle in a potential: $\displaystyle S[q] = \int_{t_1}^{t_2} \left( \frac{m}{2}\dot q^2 - V(q) \right) dt$.
• Evaluation at a point: $\delta_{x_0}[f] = f(x_0)$. This functional is linear and does not have the integral form.
• A quadratic functional: $F[f] = \int f(x)^2\, dx$.

Functional derivative

For an ordinary function of several variables, $df = \sum_i \frac{\partial f}{\partial x_i}\, dx_i$. The functional derivative $\frac{\delta F}{\delta f(x)}$ is defined in the same way, with the sum over $i$ replaced by an integral over $x$:

$$\left.\frac{d}{d\varepsilon} F[f + \varepsilon\eta]\right|_{\varepsilon=0} = \int \frac{\delta F}{\delta f(x)}\, \eta(x)\, dx \quad \text{for all test functions } \eta .$$

For $F[f] = \int f(x)^2\, dx$ we get $\frac{d}{d\varepsilon}\int (f + \varepsilon\eta)^2 dx\big|_{\varepsilon=0} = \int 2f(x)\eta(x)\, dx$, so $\frac{\delta F}{\delta f(x)} = 2f(x)$. For the evaluation functional, $\frac{\delta f(x_0)}{\delta f(x)} = \delta(x - x_0)$, the Dirac delta.

For a functional of the integral form $S[q] = \int_a^b L(q, q', x)\, dx$ and variations $\eta$ that vanish at $a$ and $b$, an integration by parts gives

$$\frac{\delta S}{\delta q(x)} = \frac{\partial L}{\partial q} - \frac{d}{dx}\frac{\partial L}{\partial q'} .$$

Setting this to zero gives the Euler-Lagrange equation, see the variational problem. For the length functional, $\partial L/\partial y = 0$ and $\partial L/\partial y' = y'/\sqrt{1+y'^2}$, so the stationary curves have constant slope: straight lines.

In field theory the same notation is used for functionals of fields $\phi(x)$ on spacetime, for example $\frac{\delta}{\delta \phi(x)}\int d^4y\, \frac{1}{2}\phi(y)^2 = \phi(x)$.

Abstract

In general, a functional is a map $F: V \to \mathbb{R}$ (or $\mathbb{C}$) on a space $V$ of functions, usually a topological vector space or an infinite-dimensional manifold. The linear continuous functionals form the dual space $V^*$. On a Hilbert space the Riesz representation theorem identifies every continuous linear functional with an inner product $F[f] = \langle g, f \rangle$. Distributions are by definition the continuous linear functionals on a space of test functions; the evaluation functional is the Dirac distribution.

For nonlinear $F$ there are two standard notions of derivative. The Gateaux derivative is the directional derivative $DF[f](\eta) = \frac{d}{d\varepsilon}F[f + \varepsilon\eta]|_{\varepsilon = 0}$. The Fréchet derivative is a bounded linear map $DF[f]$ with $F[f + h] = F[f] + DF[f](h) + o(\|h\|)$. Fréchet differentiability implies Gateaux differentiability, and the converse holds when the Gateaux derivative exists near $f$, is linear and bounded, and depends continuously on $f$ in the operator norm. The functional derivative $\delta F/\delta f(x)$ of the Concrete layer is the integral kernel of $DF[f]$, which in general is a distribution in $x$.

The functionals of the integral form above are the local functionals. Let $E \to N$ be a fibre bundle whose sections are the fields (for a path, $N = [a,b]$ and the fibre is the configuration space $\mathcal{Q}$). A Lagrangian of order $k$ is a function on the jet bundle $J^k E$, whose points record values of a section and its derivatives up to order $k$ at a point. A section $q$ has a canonical lift, the jet prolongation $j^k q: N \to J^k E$, and the local functional is

$$S[q] = \int_N (j^k q)^* \mathcal{L} ,$$

with $\mathcal{L}$ the Lagrangian density (a horizontal top form). This is the construction written as $L \circ \Gamma q$ at the top of this page. Locality makes the variational calculus tractable: the first variation of a local functional is again given by a local expression, the Euler-Lagrange form, plus boundary terms.

Contributing authors:

Leads to: Variational Calculus.

Authors: IIQ0F, Jakob Schwichtenberg