Timeline for Functional derivative commutes with total derivative
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2020 at 14:13 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
Minor formatting
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| Apr 8, 2019 at 1:03 | comment | added | user267839 | so the tool here is "to apply to test functions"? | |
| Apr 8, 2019 at 1:01 | vote | accept | user267839 | ||
| Apr 7, 2019 at 10:58 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
Added explanation
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| Apr 7, 2019 at 10:57 | comment | added | Qmechanic♦ | For starters, eq. (2') applies to functions, not functionals. | |
| Apr 7, 2019 at 2:26 | comment | added | user267839 | So this contradicts (2') or do I have overseen something? Futhermore - assume that my calculations are just wrong - how to verify that (2') holds in general? | |
| Apr 7, 2019 at 2:26 | comment | added | user267839 | I'm not sure which "simple cases" you have in mind but I think that we can take for example the "evaluation functional" $ F_{t_0}: M \to R, f \mapsto f(x_0)$ at $t_0$. Then the right hand side of (*) is $\phi(x_0)$ and on the left side we are looking for a $\frac{\delta F}{\delta\rho}(t) $ such that the integral equals $\phi(t_0)$. We see that $\frac{\delta F}{\delta\rho}(t):= \delta(t -t_0) $ solves the problem. But $\frac{\delta F}{\delta\rho(t')}(t) \neq \delta(t -t_0) \frac{\partial}{\partial \rho(t)} F$. | |
| Apr 7, 2019 at 2:26 | comment | added | user267839 | the only applyable tool to calculate $\frac{\delta}{\delta q^j(t^{\prime})}$ I found at your linked wiki page en.wikipedia.org/wiki/Functional_derivative. Here it is stated that for every functional $F\colon M \rightarrow \mathbb{R} $ by definition holds (*)$\begin{align} \int \frac{\delta F}{\delta\rho}(x) \phi(x) \; dx &= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\&= \left [ \frac{d}{d\epsilon}F[\rho+\epsilon \phi]\right ]_{\epsilon=0},\end{align}$ | |
| Apr 6, 2019 at 20:40 | comment | added | user267839 | Hi. Thank you for your enlightening explanations. One point is unclear: how do you obtain the expression $(2')$ for $\frac{\delta}{\delta q^j(t^{\prime})} $? | |
| Apr 6, 2019 at 11:33 | history | answered | Qmechanic♦ | CC BY-SA 4.0 |