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수학:범함수 [2020/01/09 17:23] – [감쇠 오일러 방정식의 분석] admin | 수학:범함수 [2020/01/10 11:40] – [세 번째 항] admin | ||
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- \int \rho(\vec{r}' | - \int \rho(\vec{r}' | ||
\right\}\\ | \right\}\\ | ||
- | &=& | + | &\approx& |
\lim_{\epsilon \to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}' | \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}' | ||
\left[ \int^{\rho(\vec{r}' | \left[ \int^{\rho(\vec{r}' | ||
Line 75: | Line 75: | ||
\end{eqnarray*} | \end{eqnarray*} | ||
+ | ====두 번째 항==== | ||
+ | \begin{eqnarray*} | ||
+ | \frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}' | ||
+ | &=& \frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}' | ||
+ | &=& \lim_{\epsilon\to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}' | ||
+ | \left[ \rho(\vec{r}'' | ||
+ | - \int \rho(\vec{r}' | ||
+ | \right\}\\ | ||
+ | & | ||
+ | \int \rho(\vec{r}' | ||
+ | +\int \rho(\vec{r}'' | ||
+ | &=& \int \rho(\vec{r}' | ||
+ | &=& 2\Phi(\vec{r}) | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | ====세 번째 항==== | ||
+ | 이 계산이 가장 간단하다. | ||
+ | \begin{eqnarray*} | ||
+ | \frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}' | ||
+ | \lim_{\epsilon\to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}' | ||
+ | - \int \rho(\vec{r}' | ||
+ | &=& | ||
+ | \int \delta (\vec{r}-\vec{r}' | ||
+ | &=& \frac{|\vec{u}(\vec{r})|^2}{2} | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | ====$F$의 미분==== | ||
+ | 연쇄법칙(chain rule)에 의해 | ||
+ | \[ \frac{dF}{dt} = \int d\vec{r} \left( \frac{\delta F}{\delta \rho(\vec{r})} \frac{\partial \rho(\vec{r}, | ||
+ | + \frac{\delta F}{\delta u_x(\vec{r})} \frac{\partial u_x(\vec{r}, | ||
+ | + \frac{\delta F}{\delta u_y(\vec{r})} \frac{\partial u_y(\vec{r}, | ||
+ | + \frac{\delta F}{\delta u_z(\vec{r})} \frac{\partial u_z(\vec{r}, | ||
+ | \right) \] | ||
+ | $\vec{u}$의 성분별로도 변분하는 과정이 필요하다. 예를 들어 | ||
+ | \begin{eqnarray*} | ||
+ | \frac{\delta}{\delta u_x(\vec{r})} \int \rho(\vec{r}' | ||
+ | \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int \rho(\vec{r}' | ||
+ | & | ||
+ | \end{eqnarray*} | ||
======참고 문헌====== | ======참고 문헌====== | ||
* T. Lancaster and S. J. Blundell, //Quantum Field Theory for the Gited Amateur// (Oxford Univerty Press, 2014). | * T. Lancaster and S. J. Blundell, //Quantum Field Theory for the Gited Amateur// (Oxford Univerty Press, 2014). | ||
* P. H. Chavanis, Eur. Phys. J. B 62, 179 (2008) [[doi: | * P. H. Chavanis, Eur. Phys. J. B 62, 179 (2008) [[doi: | ||