기록

7. Delta Notation $\frac{\partial J}{\partial \alpha }d\alpha =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\frac{\partial y}{\partial \alpha }d\alpha dx$ $\Rightarrow \delta J=\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\delta y dx$ ($ \frac{\partial J}{..

6. with Auxiliary Conditions $g=\sum_{i}x_{i}^{2}-\rho ^{2}=0 \rightarrow r=\rho =constant$: Equations of Constraint (구면 위를 따라간다) $f=f(y_{i},y_{i}';x)=f(y,y',z,z';x)$ $g(y_{i};x)=g(y,z;x)=0$ $\Rightarrow \frac{\partial y}{\partial \alpha }, \frac{\partial z}{\partial \alpha }$ are not independent (하나의 variable이 바뀌면, 다른 variable도 바뀐다) $\therefore$ for $ \alpha =0, \frac{\partial y}{\par..

5. Functions with Several Dependent Variables $f=f(y_{1}(x),y_{1}'(x),y_{2}(x),y_{2}'(x), ... ;x)$ $\Rightarrow f=f(y_{i}(x),y_{i}'(x);x), i=1,2,...,m$ $y_{i}(\alpha ,x)=y_{i}(0,x)+\alpha \eta _{i}(x)$ $\frac{\partial J}{\partial \alpha }=\int_{x_{1}}^{x_{2}}\sum_{i}(\frac{\partial f}{\partial y_{i}}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y_{i}'})\eta _{i}(x) dx=0$ $ ..

4. 2nd Form of the Euler Equation to get better form of equation for $\frac{\partial f}{\partial x}=0$ $df=\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial y'}dy'+\frac{\partial f}{\partial x}dx$ $ \frac{\mathrm{d} f}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}f(y,y';x)=\frac{\partial f}{\partial y}\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{\partial f}{\partial y'}\frac{\mathrm{d} ..

2. Euler's Equation $J=\int_{x_{1}}^{x_{2}}f(y(\alpha ,x),y'(\alpha ,x);x)dx$ $\frac{\partial J}{\partial \alpha }=\frac{\partial }{\partial \alpha }\int_{x_{1}}^{x_{2}}f(y,y';x)dx$ $=\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha }+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial \alpha })dx$ ($ \because y(\alpha ,x)=y(0,x)+\alpha \eta (x) \ri..

Calculus of Variation 카테고리는 5th (Marion)의 chapter 6 순서를 따라갑니다 1. Calculus of Variation find an extreme solution (max or min) $J=\int_{x_{1}}^{x_{2}}f(y(x),y'(x);x)dx$ ($y'\equiv \frac{\mathrm{d} y}{\mathrm{d} x}$ ) (x: independent variable, y: dependent variable) $y=y(\alpha ,x)$ $\alpha =0 \rightarrow y=y(0,x)=y(x)$: function that yields an extreme for J $ \Rightarrow y=y(\alpha ,x)=y(0..