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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)…

Contributing authors:

IIQ0F Jakob Schwichtenberg
basic_tools/variational_calculus/functional.txt · Last modified: 2020/04/12 14:41 by jakobadmin