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I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x)$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x)}{\partial \alpha}$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to Y (the wrong function) and not y(x)?

The book I'm referring to is Classical mechanics by John R. Taylor

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{6.6}$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x).$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x)}{\partial \alpha}.$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to $Y$ (the wrong function) and not $y(x)$?

The book I'm referring to is Classical mechanics by John R. Taylor

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x)$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x)}{\partial \alpha}$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to Y (the wrong function) and not y(x)?

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x)$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x)}{\partial \alpha}$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to Y (the wrong function) and not y(x)?

The book I'm referring to is Classical mechanics by John R. Taylor

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x)$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x}{\partial \alpha}$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to Y (the wrong function) and not y(x)?

I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)$$ Where $y(x)$ is the minimal path and $\eta(x)$ is some arbitrary function. He then argues that since $y(x)$ is the minimum any other function, no matter how close to $y(x)$ must be greater than $y(x)$. So using this fact, he parameterizes the action integral $S$ by $\alpha$. That is, $$S(\alpha)= f(y+\alpha \eta, y' + \alpha \eta, x)$$

My confusion is that, later in the chapter he says this the Euler-Lagrange Equation finds the stationary point not the minimum but the derivation used the fact that for all $\alpha$ other than zero, $y<Y$. How does that work?

Secondly, in order for $S(\alpha)$ to be stationary he puts, $$\frac{dS}{d\alpha} = \int \frac{\partial f(y+\alpha \eta, y'+\alpha \eta ', x)}{\partial \alpha}$$ Here since $\alpha$ appears in two arguments of $f$, the derivative will be (using chain rule): $$\frac{\partial S}{\partial \alpha} = \eta \frac{\partial f}{\partial y} + \eta' \frac{\partial f}{\partial y'}$$ I don't understand this step. How do you know you'll have two terms? And also where did that addition sign came from? Also shouldn't it be the partial with respect to Y (the wrong function) and not y(x)?