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--- basic_tools:variational_calculus:functional [2018/03/10 17:28]
iiqof Clarification
+++ basic_tools:variational_calculus:functional [2018/03/15 14:31]
iiqof
@@ Line 2: / Line 2: @@
-Let $\Omega(\mathcal{Q})$ be the set of functions $q:\mathbb{R} \to \mathcal{Q}$, then a //functional// F is a map
+Let $\Omega(\mathcal{Q})$ be the set of functions $q:\mathbb{R^n} \to \mathcal{Q}$, then a //functional// S is a map
 $$
-F:\Omega \to \mathbb{R}; F[q] \mapsto \alpha \in\mathbb{R}
+S:\Omega \to \mathbb{R}; S[q] \mapsto \alpha \in\mathbb{R}
 $$
-So we can see how a functional is a //function of functions// as we said before, this is the reason why the notation $F[\cdot]$ instead of $F(\cdot)$, to remind that it is more that the eyes meet.
+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.
@@ Line 18: / Line 18: @@
 $$
-S[q] = \int_a^b F(q(x),  q'(x),  q''(x), \dots, x)d x
+S[q] = \int_a^b L(q(x),  q'(x),  q''(x), \dots, x)d x
 $$
-with $F: T^{(n)}\mathcal Q \times [a,b] \to \mathbb R$. Note that $F$ is a function from a manifold to the reals. And what is integrated is $F\circ \Gamma q(x)$, where $\Gamma$ is the [[::|lift]] of the function $q$ to its [[::|fibres]].
+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]].
@@ Line 27: / Line 27: @@
-But there can be other functionals: maximum/minimum value of a function, value at point $x$...
+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)...

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