Post History
#2: Post edited
- $$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{x}}\frac{\partial \dot{x}}{\partial \alpha})\mathrm dx$$
- Consider the second of these integrals:
- $$\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx=\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x \partial \alpha}\mathrm dx$$
What did they mean by "second"? There must be a negative on LHS or RHS. What happened to the line? To me, it seems like he had used chain rule for $\partial \dot{y}$. But, why there's no negative?- >> <span class="hide">used double blockquote to decrease size of the picture</span>
- $$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{x}}\frac{\partial \dot{x}}{\partial \alpha})\mathrm dx$$
- Consider the second of these integrals:
- $$\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx=\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x \partial \alpha}\mathrm dx$$
- What did they mean by "second"? There must be a negative on LHS or RHS. What happened to the line? To me, it seems like he had used chain rule for $\partial y$. But, why there's no negative?
- >> <span class="hide">used double blockquote to decrease size of the picture</span>
#1: Initial revision
Consider the second of these integrals (What's the meaning of second right here?)
$$\frac{dJ}{d\alpha}=\int_{x_1}^{x_2}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{x}}\frac{\partial \dot{x}}{\partial \alpha})\mathrm dx$$
Consider the second of these integrals:
$$\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\mathrm dx=\int_{x_1}^{x_2}\frac{\partial f}{\partial \dot{y}}\frac{\partial^2 y}{\partial x \partial \alpha}\mathrm dx$$
What did they mean by "second"? There must be a negative on LHS or RHS. What happened to the line? To me, it seems like he had used chain rule for $\partial \dot{y}$. But, why there's no negative?
>> <span class="hide">used double blockquote to decrease size of the picture</span>