Consider the second of these integrals (What's the meaning of second right here?)

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$\frac{d J}{d α} = \int_{x_{1}}^{x_{2}} (\frac{\partial f}{\partial y} \frac{\partial y}{\partial α} + \frac{\partial f}{\partial \dot{x}} \frac{\partial \dot{x}}{\partial α}) d x$ Consider the second of these integrals: $\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial \dot{y}} \frac{\partial \dot{y}}{\partial α} d x = \int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial \dot{y}} \frac{\partial^{2} y}{\partial x \partial α} d x$

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?

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What did they mean by "second"?

They've mentally expanded $\frac{d J}{d α} = \int_{x_{1}}^{x_{2}} (\frac{\partial f}{\partial y} \frac{\partial y}{\partial α} + \frac{\partial f}{\partial \dot{y}} \frac{\partial \dot{y}}{\partial α}) d x$ (which I've corrected to be what it says in the image rather than your MathJax) as $\frac{d J}{d α} = \int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial y} \frac{\partial y}{\partial α} d x + \int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial \dot{y}} \frac{\partial \dot{y}}{\partial α} d x$

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Peter Taylor‭

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I wasn't talking about this one (2 comments)

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In the post, actually $\dot{y} = \frac{d y}{d x}$ . So,

$\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial \dot{y}} \frac{\partial \dot{y}}{\partial α} d x = \int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial \dot{y}} \frac{\partial^{2} y}{\partial x \partial α} d x$

In the equation, they just wrote $\frac{d y}{d x}$ instead of $\dot{y}$

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Yes, that's what I said in my comment yesterday. (2 comments)

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Consider the second of these integrals (What's the meaning of second right here?)

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