basic_tools:variational_calculus:functional
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basic_tools:variational_calculus:functional [2018/03/10 17:05] iiqof created Functional Page |
basic_tools:variational_calculus:functional [2020/04/12 14:41] (current) jakobadmin |
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| - | 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. | + | 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. |
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| $$ | $$ | ||
| - | 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]]. |
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| - | 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)... |
basic_tools/variational_calculus/functional.1520697935.txt.gz · Last modified: 2018/03/10 16:05 (external edit)
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