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故有,$\left\{ {\begin{array}\ {x=c_1sint,} \\ {y=-c_1cost+c_2} \\ \end{array} } \right.$\\ | 故有,$\left\{ {\begin{array}\ {x=c_1sint,} \\ {y=-c_1cost+c_2} \\ \end{array} } \right.$\\ | ||
消去参数t,得,$x^2+(y-c_2)^2=c_1^2$.\\ | 消去参数t,得,$x^2+(y-c_2)^2=c_1^2$.\\ | ||
- | 这表示圆心在纵轴上的一簇圆,其中,$c_1$,$c_2$由边界条件确定。 | + | 这表示圆心在纵轴上的一簇圆,其中,$c_1$,$c_2$由边界条件确定。 \\ |
- | <note important>edited by 杨冰(0921060)2010/05/22 </note> | + | <note important>edited by 杨冰(0921060)2010/05/22 </note> \\ |
∗**例7.** 求最速降线问题的解,即求 \\ $T=\int_{0}^{a} {\frac{\sqrt {1+y'^2}}{\sqrt{2gy}}}dx$,$y(0)=0$,$y(a)=b$,的极值曲线。\\ | ∗**例7.** 求最速降线问题的解,即求 \\ $T=\int_{0}^{a} {\frac{\sqrt {1+y'^2}}{\sqrt{2gy}}}dx$,$y(0)=0$,$y(a)=b$,的极值曲线。\\ | ||
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由t=0时,x=0,得$c_2=0$,而$c_1$由y(a)=b定出。于是,最速降线问题的解为\\ | 由t=0时,x=0,得$c_2=0$,而$c_1$由y(a)=b定出。于是,最速降线问题的解为\\ | ||
$\left\{ {\begin{array}\ {x=\frac{c_1}{2}(t-sint),} \\ {y= \frac{c_1}{2}\frac{c_1}{2}(1-cost)}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {x=\frac{c_1}{2}(t-sint),} \\ {y= \frac{c_1}{2}\frac{c_1}{2}(1-cost)}. \\ \end{array} } \right.$\\ | ||
- | 摆线。这是一条过A,B两点的摆线。 | + | 摆线。这是一条过A,B两点的摆线。\\ |
∗**例8.** 求最小旋转面面积问题的解,即求泛函 \\ | ∗**例8.** 求最小旋转面面积问题的解,即求泛函 \\ | ||
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$\left\{ {\begin{array}\ {x={c_1}t-c_2,} \\ {y= c_1cosht}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {x={c_1}t-c_2,} \\ {y= c_1cosht}. \\ \end{array} } \right.$\\ | ||
绕OX轴旋转而成的,消去参数t,得 \\ | 绕OX轴旋转而成的,消去参数t,得 \\ | ||
- | $y=c_1cosh\frac{x-c_2}{c_1}, \\ | + | $y=c_1cosh\frac{x-c_2}{c_1},$ \\ |
- | 其中,$c_1$,$c_2$由$y(x_0)=y_0$,$y(x_1)=y_1$确定,因此,最小曲面是悬链面。 | + | 其中,$c_1$,$c_2$由$y(x_0)=y_0$,$y(x_1)=y_1$确定,因此,最小曲面是悬链面。 \\ |
- | + | ====**2.2. 含有多个未知函数的变分问题**==== \\ | |
- | ===== **2.2. 含有多个未知函数的变分问题** ===== | + | ★含两个未知函数的泛函极值的必要条件 \\ |
- | ★含两个未知函数的泛函极值的必要条件 | + | |
☆问题: \\ | ☆问题: \\ | ||
$J[y(x),z(x)]=\int_{x_0}^{x_1}{F(x,y,z,y',z')}dx,$ (11) \\ | $J[y(x),z(x)]=\int_{x_0}^{x_1}{F(x,y,z,y',z')}dx,$ (11) \\ | ||
其中,F关于所含变量具有二阶连续领导,容许曲线$y(x)$,$z(x)$∈$C^2[x_0,x_1]$,且满足边界条件: \\ | 其中,F关于所含变量具有二阶连续领导,容许曲线$y(x)$,$z(x)$∈$C^2[x_0,x_1]$,且满足边界条件: \\ | ||
- | $\left\{ {\begin{array}\ {y(x_0)=y_0, y(x_1)=y_1,} \\ {z(x_0)=z_0, z(x_1)=z_1}. \\ \end{array} } \right.$\\ | + | $\left\{ {\begin{array}\ {y(x_0)=y_0, y(x_1)=y_1,} \\ {z(x_0)=z_0, z(x_1)=z_1}. \\ \end{array} } \right.$ \\ |
☆极值的必要条件:使泛函$J[y(x),z(x)]=\int_{x_0}^{x_1}{F(x,y,z,y',z')}dx$达到极值的必要条件是,y(x),z(x)满足Eular方程组,即: \\ | ☆极值的必要条件:使泛函$J[y(x),z(x)]=\int_{x_0}^{x_1}{F(x,y,z,y',z')}dx$达到极值的必要条件是,y(x),z(x)满足Eular方程组,即: \\ | ||
- | $\left\{ {\begin{array}\ {F_y-\frac{d}{dx}F_{y'}=0,} \\ {F_z-\frac{d}{dx}F_{z'}=0}. (12) \\ \end{array} } \right.$\\ | + | $\left\{ {\begin{array}\ {F_y-\frac{d}{dx}F_{y'}=0,} \\ {F_z-\frac{d}{dx}F_{z'}=0}. \\ \end{array} } \right.$ (12)\\ |
- | ☆例1.求泛函$J[y,z]=\int_{0}^{\frac{π}{2}{(y'^2+z'^2+2yz)}dx$满足边界条件: \\ | + | ☆例1.求泛函$J[y,z]=\int_{0}^{\frac{π}{2}{(y'^2+z'^2+2yz)}}dx$满足边界条件: \\ |
- | $\left\{ {\begin{array}\ {y(x_0)=0, y(\frac{π}{2})=1,} \\ {z(x_0)=0, z(\frac{π}{2})=-1}. \\ \end{array} } \right.$\\ | + | $\left\{ {\begin{array}\ {y(x_0)=0, y(\frac{π}{2})=1,} \\ {z(x_0)=0, z(\frac{π}{2})=-1}. \\ \end{array} } \right.$ \\ |
的极值曲线。 \\ | 的极值曲线。 \\ | ||
解:$F=y'^2+z'^2+2yz$,故Eular方程组为:\\ | 解:$F=y'^2+z'^2+2yz$,故Eular方程组为:\\ | ||
- | $\left\{ {\begin{array}\ {2z-\frac{d}{dx}(2y'}=0,} \\ {2y-\frac{d}{dx}(2z')=0}. \\ \end{array} } \right.$\\ | + | $\left\{ {\begin{array}\ {2z-\frac{d}{dx}(2y')=0,} \\ {2y-\frac{d}{dx}(2z')=0}. \\ \end{array} } \right.$\\ |
得: \\ | 得: \\ | ||
- | $\left\{ {\begin{array}\ {z-y''=0,} \\ {y-z''=0}. \\ \end{array} } \right.$\\ | + | $\left\{ {\begin{array}\ {z-y^{(2)}=0,} \\ {y-z^{(2)}=0}. \\ \end{array} } \right.$\\ |
由此方程组消去z,得$y^{(4)}-y=0$,其通解为:\\ | 由此方程组消去z,得$y^{(4)}-y=0$,其通解为:\\ | ||
$\left\{ {\begin{array}\ {y=c_1e^x+c_2e^{-x}+c_3cosx+c_4sinx,} \\ {z=y''=c_1e^x+c_2e^{-x}-c_3cosx-c_4sinx}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {y=c_1e^x+c_2e^{-x}+c_3cosx+c_4sinx,} \\ {z=y''=c_1e^x+c_2e^{-x}-c_3cosx-c_4sinx}. \\ \end{array} } \right.$\\ | ||
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$\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | $\left\{ {\begin{array}\ {y=sinx,} \\ {z=-sinx}. \\ \end{array} } \right.$\\ | ||
- | 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> |