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--- 수학:범함수 [2020/01/09 17:07] – [예] admin
+++ 수학:범함수 [2020/01/10 11:40] – [세 번째 항] admin
@@ Line 38: / Line 38: @@
 을 얻게 된다.
-=====감쇠 오일러 방정식의 분석=====
+=====또다른 예: 감쇠 오일러 방정식의 분석=====
 밀도장 $\rho(\vec{r},t)$와 퍼텐셜 $U(|\vec{r}-\vec{r}'|)$에 대해 $\Phi(\vec{r},t) \equiv \int \rho(\vec{r}',t) U(|\vec{r}-\vec{r}'|) d\vec{r}'$라고 정의하자. 다음과 같은 계를 고려하는데
 \begin{eqnarray*}
@@ Line 52: / Line 52: @@
 임을 보일 수 있다.
+====첫 번째 항====
+표기를 약간 간단하게 하기 위해 $\Psi(\rho) \equiv \int^\rho \frac{P(\rho')}{\rho'^2} d\rho'$라 하자.
+\begin{eqnarray*}
+\frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}') \Psi(\vec{r}') d\vec{r}'&=&
+\lim_{\epsilon \to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta (\vec{r}'-\vec{r}) \right]
+\int^{\rho(\vec{r}') + \epsilon \delta (\vec{r}'-\vec{r})} \frac{P(\rho')}{\rho'^2} d\rho' d\vec{r}'
+- \int \rho(\vec{r}') \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho' d\vec{r}'
+\right\}\\
+&\approx&
+\lim_{\epsilon \to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta (\vec{r}'-\vec{r}) \right]
+\left[ \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho'
+ + \frac{P[\rho(\vec{r}')]}{\rho^2(\vec{r}')} \epsilon \delta (\vec{r}'-\vec{r})
+\right] d\vec{r}'
+- \int \rho(\vec{r}') \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho' d\vec{r}'
+\right\}\\
+&\approx&
+\int \rho(\vec{r}') \frac{P[\rho(\vec{r}')]}{\rho^2(\vec{r}')} \delta(\vec{r}'-\vec{r}) d\vec{r}' + \int \delta(\vec{r}'-\vec{r}) \int^{\rho(\vec{r}')} \frac{P(\rho')}{\rho'^2} d\rho'\\
+&=&
+\frac{P[\rho(\vec{r})]}{\rho(\vec{r})} + \int^{\rho(\vec{r})} \frac{P(\rho')}{\rho'^2} d\rho'\\
+&=&
+\frac{P[\rho(\vec{r})]}{\rho(\vec{r})} + \Psi[\rho(\vec{r})]
+\end{eqnarray*}
+====두 번째 항====
+\begin{eqnarray*}
+\frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}') \Phi(\vec{r}') d\vec{r}'
+&=& \frac{\delta}{\delta \rho(\vec{r})} \int \rho(\vec{r}') \int \rho(\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'\\
+&=& \lim_{\epsilon\to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta(\vec{r}-\vec{r}') \right]
+\left[ \rho(\vec{r}'') + \epsilon \delta(\vec{r}-\vec{r}'') \right] U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
+- \int \rho(\vec{r}') \int \rho(\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
+\right\}\\
+&\approx&
+\int \rho(\vec{r}') \delta(\vec{r}-\vec{r}'') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'
++\int \rho(\vec{r}'') \delta(\vec{r}-\vec{r}') U(|\vec{r}'-\vec{r}''|) d\vec{r}'' d\vec{r}'\\
+&=& \int \rho(\vec{r}') U(|\vec{r}'-\vec{r}|) d\vec{r}' + \int \rho(\vec{r}'') U(|\vec{r}''-\vec{r}|) d\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}') \frac{|\vec{u}(\vec{r}')|^2}{2} d\vec{r}' &=&
+\lim_{\epsilon\to 0} \frac{1}{\epsilon} \left\{ \int \left[ \rho(\vec{r}') + \epsilon \delta (\vec{r}-\vec{r}') \right] \frac{|\vec{u}(\vec{r}')|^2}{2} d\vec{r}'
+- \int \rho(\vec{r}') \frac{|\vec{u}(\vec{r}')|^2}{2} d\vec{r}'\right\}\\
+&=&
+\int \delta (\vec{r}-\vec{r}') \frac{|\vec{u}(\vec{r}')|^2}{2} d\vec{r}'\\
+&=& \frac{|\vec{u}(\vec{r})|^2}{2}
+\end{eqnarray*}
+====$F$의 미분====
+연쇄법칙(chain rule)에 의해
+\[ \frac{dF}{dt} = \int d\vec{r} \left( \frac{\delta F}{\delta \rho(\vec{r})} \frac{\partial \rho(\vec{r},t)}{\partial t}
++ \frac{\delta F}{\delta u_x(\vec{r})} \frac{\partial u_x(\vec{r},t)}{\partial t}
++ \frac{\delta F}{\delta u_y(\vec{r})} \frac{\partial u_y(\vec{r},t)}{\partial t}
++ \frac{\delta F}{\delta u_z(\vec{r})} \frac{\partial u_z(\vec{r},t)}{\partial t}
+\right) \]
+$\vec{u}$의 성분별로도 변분하는 과정이 필요하다. 예를 들어
+\begin{eqnarray*}
+\frac{\delta}{\delta u_x(\vec{r})} \int \rho(\vec{r}') \frac{u_x^2(\vec{r}') + u_y^2(\vec{r}') + u_z^2(\vec{r}')}{2} d\vec{r}' &=&
+\lim_{\epsilon \to 0} \frac{1}{\epsilon} \int \rho(\vec{r}') \frac{[u_x(\vec{r}') + \epsilon \delta(\vec{r}'-\vec{r})]^2 - u_x^2(\vec{r}')}{2} d\vec{r}'\\
+&\approx& \int \rho(\vec{r}') \delta(\vec{r}'-\vec{r}) u_x(\vec{r}') d\vec{r}' = \rho(\vec{r}) u_x(\vec{r})
+\end{eqnarray*}
 ======참고 문헌======
   * 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:10.1140/epjb/e2008-00142-9|(link)]].