端点变化的变分问题

2023-07-23 15:58 作者:艾琳娜的糖果屋 0人读过 | 我要投稿

考虑泛函 $%0AJ%5Cleft%5B%20y%20%5Cright%5D%20%3D%5Cint_%7Bx_1%7D%5E%7Bx_2%7D%7BF%5Cleft(%20x%2Cy%2Cy%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%0A%0A$ 左端点固定 $%0Ay%5Cleft(%20x_1%20%5Cright)%20%3Dy_1%0A%0A$ ，右端点在 $y%3D%5Cvarphi%20%5Cleft(%20x%20%5Cright)%20$ 上待定。

我们先假设J在y处取得极值，那么对于一个小扰动 $%0A%5Cvarepsilon%20%5Ceta(x)%20%0A%0A$ ,右端点会随着扰动的变化而变化，也就是说此时 $%0Ax_2%3Dx_2%5Cleft(%20%5Cvarepsilon%20%5Cright)%20%2Cx_2%5Cleft(%200%20%5Cright)%20%3Dx_2%0A%0A$ $%0A%0A%0A$

而在右端点处满足 $%0Ay%5Cleft(%20x_2%5Cleft(%20%5Cvarepsilon%20%5Cright)%20%5Cright)%20%2B%5Cvarepsilon%20%5Ceta%20%5Cleft(%20x_2%5Cleft(%20%5Cvarepsilon%20%5Cright)%20%5Cright)%20%3D%5Cvarphi%20%5Cleft(%20x_2%5Cleft(%20%5Cvarepsilon%20%5Cright)%20%5Cright)%20%0A%0A$ ，从而解得

$%0Ax_2%5Cprime%5Cleft(%200%20%5Cright)%20%3D%5Cfrac%7B%5Ceta%20%5Cleft(%20x_2%20%5Cright)%7D%7B%5Cvarphi%20%5Cprime%5Cleft(%20x_2%20%5Cright)%20-y%5Cprime%5Cleft(%20x_2%20%5Cright)%7D%0A%0A$

$%0A%5Cdelta%20J%3D%5Cfrac%7B%5Cmathrm%7Bd%7DJ%5Cleft%5B%20y%2B%5Cvarepsilon%20%5Ceta%20%5Cright%5D%7D%7B%5Cmathrm%7Bd%7D%5Cvarepsilon%7D%5Cmid_%7B%5Cvarepsilon%20%3D0%7D%5E%7B%7D%0A%0A%3D%5Cint_%7Bx_1%7D%5E%7Bx_2%7D%7B%5Cleft(%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%7D%5Ceta%20%2B%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%5Cprime%7D%5Ceta%20%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%2BF%5Cleft(%20x%2Cy%2Cy%5Cprime%20%5Cright)%20%5Cmid_%7Bx%3Dx_2%7D%5E%7B%7Dx_2%5Cprime%5Cleft(%200%20%5Cright)%20%0A$

$%0A%3D%5Cint_%7Bx_1%7D%5E%7Bx_2%7D%7B%5Cleft(%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%7D-%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dx%7D%5Cleft(%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%5Cprime%7D%20%5Cright)%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%2B%0A%5Cleft(%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%5Cprime%7D%2B%5Cfrac%7BF%7D%7B%5Cvarphi%20%5Cprime%5Cleft(%20x%20%5Cright)%20-y%5Cprime%5Cleft(%20x%20%5Cright)%7D%20%5Cright)%20%5Ceta%20%5Cmid_%7Bx%3Dx_2%7D%5E%7B%7D%3D0%0A%0A%0A%0A$

由于y是极值函数，当右端点确定下来后它必然满足边界固定的欧拉-拉格朗日方程，因此第一项等于0，而 $%0A%5C%2C%5C%2C%5Ceta%20%5Cleft(%20x_2%20%5Cright)%20%5Cne%200%0A%0A%0A%0A%0A$

由此我们得到了横截条件 $%0AF%2BF_%7By%5Cprime%7D%5Cleft(%20%5Cvarphi%20%5Cprime%5Cleft(%20x%20%5Cright)%20-y%5Cprime%5Cleft(%20x%20%5Cright)%20%5Cright)%20%5Cmid_%7Bx%3Dx_2%7D%5E%7B%7D%3D0%0A%0A$

同理可以得到两端都变化的泛函 $%0AJ%5Cleft%5B%20y%20%5Cright%5D%20%3D%5Cint_%7Bx_1%7D%5E%7Bx_2%7D%7BF%5Cleft(%20x%2Cy%2Cy%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%0A%0A$ ，左端点在 $%0Ay%3D%5Cpsi%20%5Cleft(%20x%20%5Cright)%20%0A%0A$ 上移动，右端点在 $y%3D%5Cvarphi%20%5Cleft(%20x%20%5Cright)%20$ 上移动

则有横截条件 $%0AF%2BF_%7By%5Cprime%7D%5Cleft(%20%5Cpsi%20%5Cprime%5Cleft(%20x%20%5Cright)%20-y%5Cprime%5Cleft(%20x%20%5Cright)%20%5Cright)%20%5Cmid_%7Bx%3Dx_1%7D%5E%7B%7D%3D0%0A%5C%5C%0A%0AF%2BF_%7By%5Cprime%7D%5Cleft(%20%5Cvarphi%20%5Cprime%5Cleft(%20x%20%5Cright)%20-y%5Cprime%5Cleft(%20x%20%5Cright)%20%5Cright)%20%5Cmid_%7Bx%3Dx_2%7D%5E%7B%7D%3D0%0A%0A$

以及欧拉-拉格朗日方程 $%0A%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%7D-%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dx%7D%5Cleft(%20%5Cfrac%7B%5Cpartial%20F%7D%7B%5Cpartial%20y%5Cprime%7D%20%5Cright)%20%3D0%0A%0A$

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