✎ Functional [The Physics Travel Guide]

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====== Functional ======

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 the usual form that a functional takes is that of an integral of an algebraic combination of the a function and its derivaties:

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

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. a function is also a functional)...

Please solve the following equation to prove you're human. 162 -13 = Please keep this field empty:

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