keynote:lesson11
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keynote:lesson11 [2010/06/26 09:27] 10921048 |
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| $\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | ||
| - | <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 \\ | + | ★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 \\ | 解：因为F_y=F_z=0,故Euler方程组为 \\ | ||
| $\left\{ {\begin{array}\ {\frac{d}{dx}(F_{y'})=0,} \\ {\frac{d}{dx}(F_{z'})=0}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {\frac{d}{dx}(F_{y'})=0,} \\ {\frac{d}{dx}(F_{z'})=0}. \\ \end{array} } \right.$\\ | ||
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| 是所求的极值曲线。\\ | 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\\ | ·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 \\ | + | 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 \\ |
| $\left\{ {\begin{array}\ {y_i(x_0)=y_{i0},} \\ {y_i(x_1)=y_{i1}},i=1,2,…n, \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {y_i(x_0)=y_{i0},} \\ {y_i(x_1)=y_{i1}},i=1,2,…n, \\ \end{array} } \right.$\\ | ||
| 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\\ | 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\\ | ||
keynote/lesson11.txt · Last modified: 2023/08/19 21:02 (external edit)
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