Euler’s approach to variational calculus

[Pencilvester]

Hello PF, I’m going through a book called “A First Course in the Calculus of Variations.” I can’t remember who the author is at the moment, I’ll post it later. Anyway, I’m having trouble with one part: suppose we have a function $y (x)$ that gives a continuous polygonal curve from $x = a$ to $x = b$ with $n + 1$ pieces and values $y_i$ separated by $Δx = \frac {b - a} {n + 1}$ and endpoints fixed at $y_0$ and $y_{n + 1}$. We also have$$J = \sum_{i = 0} ^n f(x_i , y_i , \frac {y_{i + 1} - y_i} {Δx}) Δx$$Now we want to take the derivative of $J$ with respect to a specific $y_k$, which will appear in 2 terms of the sum
($i = k$ and $i = k - 1$). As far as I know, $f$ could be any sort of function, not necessarily linear on $x$, $y$, and $y’$
($y’ ≡ \frac {dy} {dx}$ which of course is the same as $\frac {y_{i + 1} - y_i} {Δx}$ as $Δx$ approaches 0 and $n$ approaches infinity, which is the limit we will eventually be taking). So without any explanation, the book says$$\frac {∂J} {∂y_k} = f_y (x_k , y_k , \frac {y_{k + 1} - y_k} {Δx}) Δx + f_{y’} (x_{k - 1} , y_{k - 1} , \frac {y_k - y_{k - 1}} {Δx}) - f_{y’} (x_k , y_k , \frac {y_{k + 1} - y_k} {Δx})$$(call this eq. 1) And this is what I am having trouble with. First of all, they use subscripts on $f$ that I assume are to indicate partial derivatives, but I am not certain as this is the first time in the book they use this notation, and they do not have any place in the book that gives all of their notation conventions. Anyway, the only way I can make sense of this is if, in general,$$\frac {∂} {∂y} f (x , y , g (y) ) = \frac {∂} {∂y} (f) + \frac {∂} {∂g} (f) ⋅ \frac {dg} {dy}$$(call this eq. 2) where $f$ is any function of variables $x$ and $y$ and function $g$, which is itself a function of $y$, but on the RHS of the equation we treat $g$ as just another variable (holding $g$ constant while we vary $y$ a little, and vise versa). So this is my main question: is equation 2 true in general? If so, where could I find a proof for it? If not, how do we get eq. 1? Any help would be much appreciated.

 

[andrewkirk]

I think you're on the right track. Think of $f$ as a function of three variables $x,y,y'$ and $g$ as a function of four. Then we can re-write $J$ as a function of $2(n+1)$ variables $x_0,...,x_n,y_0,...,y_n$ as follows:

$$J(x_0,...,x_n,y_0,...,y_n) = \sum_{i=0}^n f(x_i,y_i,g(x_i,x_{i+1},y_i,y_{i+1}))(x_{i+1}-x_i)$$
where
$$g(x_i,x_{i+1},y_i,y_{i+1}) = \frac{y_{i+1}-y_i}{x_{i+1}-x_i}$$

Then we can write $\frac{\partial J}{\partial y_k}$ the way you have above as a total differential. The first part gives us the first term, being from the sum component with $n=k$ and the second part gives us the second and third terms, from the sum components with $n\in\{k,k-1\}$.

Your interpretation of what the subscripts to $f$ mean is correct, but it is bad practice of the author to write his partial derivatives like $f_{y'}(x_k,y_k,\frac{y_{k+1}-y_k}{x_{k+1}-x_k})$ when none of the arguments are named $y'$. Better practice is to write:
$$f_{y'}(x,y,y')|_{x=x_k,y=y_k,y'=\frac{y_{k+1}-y_k}{x_{k+1}-x_k}}$$
where the subscript next to the vertical bar indicates assignments of values to the arguments of $f$, to be done after the differentiation has been performed.

 

[Pencilvester]

Then we can write $\frac{\partial J}{\partial y_k}$ the way you have above as a total differential. The first part gives us the first term, being from the sum component with $n=k$ and the second part gives us the second and third terms, from the sum components with $n\in\{k,k-1\}$.

So you’re saying my equation 2 is correct, right? You wouldn’t happen to be able to reference a proof for that equation, would you? I mean, if I were to assume $f$ was linear, then it would be clearly intuitive, but for any $f$ in general, it becomes much less intuitive to me. Also, how was my own notation on equation 2? I just found it awkward to use dels on both sides when on each side of the equation the derivative operators were technically doing different things.

Your interpretation of what the subscripts to $f$ mean is correct, but it is bad practice of the author to write his partial derivatives like $f_{y'}(x_k,y_k,\frac{y_{k+1}-y_k}{x_{k+1}-x_k})$ when none of the arguments are named $y'$.

Thank you for validating my exact thoughts. In the book, he didn’t even include any explanation linking $y’$ to $\frac {y_{i + 1} - y_i} {Δx}$. The first place $y’$ came up in this part of the book was in that equation 1 as a subscript (obviously I made the connection, it just took a few confused minutes).

 

[Pencilvester]

Hello PF, I’m going through a book called “A First Course in the Calculus of Variations.” I can’t remember who the author is at the moment, I’ll post it later.

Oh, and for everyone’s reference, the author of the book is Mark Kot.

 

[andrewkirk]

You wouldn’t happen to be able to reference a proof for that equation, would you?

The equation is called the Total Derivative Law. The wiki page on it is here. There is probably a proof either on that page or somewhere linked on that page. If not, just googling 'proof of total derivative law' should find something pretty quickly. IIRC, the proof is short and simple, using limits.

 

[Pencilvester]

The equation is called the Total Derivative Law. The wiki page on it is here. There is probably a proof either on that page or somewhere linked on that page. If not, just googling 'proof of total derivative law' should find something pretty quickly. IIRC, the proof is short and simple, using limits.

Thanks!