keynote:lesson11
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
keynote:lesson11 [2010/06/10 17:08] 10921062 |
keynote:lesson11 [2023/08/19 21:02] (current) |
||
|---|---|---|---|
| Line 129: | Line 129: | ||
| $\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | ||
| - | <note> edited by 谭小球(0921062)2010/06/10</note> | + | <note> edited by 谭小球(0921062)2010/06/10</note>\\ |
| + | ★例2 求泛函$J[y(x),z(x)]=\int_{x_0}^{x_1}{F(y',z')dx,}$ 的极值曲线，其中假设 F_{y'y'}-F^2_{y'z'} ≠0 \\ | ||
| + | 解：因为F_y=F_z=0,故Euler方程组为 \\ | ||
| + | $\left\{ {\begin{array}\ {\frac{d}{dx}(F_{y'})=0,} \\ {\frac{d}{dx}(F_{z'})=0}. \\ \end{array} } \right.$\\ | ||
| + | 得：\\ | ||
| + | $\left\{ {\begin{array}\ {F_{y'y'}y"+F_{y'z'}z"=0,} \\ {F_{y'z'}y"+F_{z'z'}z"=0}. \\ \end{array} } \right.$\\ | ||
| + | 根据假设条件，该方程只有零解y"=0,z"=0,即 \\ | ||
| + | $\left\{ {\begin{array}\ {y=c_1x+c_2,} \\ {z=c_3x+c_4}. \\ \end{array} } \right.$\\ | ||
| + | 是所求的极值曲线。\\ | ||
| + | ·含n(n>2)个未知函数的泛函极值的必要条件\\ | ||
| + | 泛函$J=\int_{x_0}^{x_1}{F(x,y_1,y_2,…,y_n,y'_1,y'_2,…y'_n)dx,}$在满足边界条件 \\ | ||
| + | $\left\{ {\begin{array}\ {y_i(x_0)=y_{i0},} \\ {y_i(x_1)=y_{i1}},i=1,2,…n, \\ \end{array} } \right.$\\ | ||
| + | 下，取得极值的必要条件是y_1(x),y_2(x),…,y_n(x),满足Euler方程组\\ | ||
| + | F_{yi}-\frac{d}{dx}(F_{y'i})=0,i=1,2,…n. \\ | ||
| + | 上述方程组通解中的常数，可以由所给的边界条件确定。\\ | ||
| + | <note> edited by 杨立春(10921048)2010/06/26</note> | ||
keynote/lesson11.1276160927.txt.gz · Last modified: 2023/08/19 21:01 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 4.0 International
