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I'm having trouble wrapping my head around the functions involved in the derivation of the Euler-Lagrange equation.

Although, as a sidenote, I'm deriving it by trying to prove that the shortest path between two points ($x_a$ and $x_b$) on a plane ($\mathbb{R}^2$), is a straight line.

I understand that a path between two points can be defined as a set of points. A variable in this set is a point $(x, f(x))$, for some $x\in\mathbb{R}$, and function $f:\mathbb{R}\rightarrow\mathbb{R}$.

I also understand that using both the Pythagorean theorem, and the fundamental theorem of calculus, the length of a path is: $L=\int_{x_a}^{x_b}\sqrt{(1+f'(x_i))} dx$.

What I don't understand is how to express $L$ as a function, specifically, what the input and ouput sets of the function are.

Here's what I'm getting at first glance. Let's first denote $\mathsf{F}$ as the set of all possible functions describing the paths joining $x_a$ and $x_b$. Then, the total length $L$ is a function $L: \mathsf{F}\rightarrow\mathbb{R}$. That is, the total length, $L$ relates a function/path to a real number representing its length.

However, firstly, books represent the function $L$ as a multivariate function with output $L(f, f', x)$. This seems unnecessarily cumbersome (why use a triple $(f, f', x)$ as an input, when just the single input $f$ will suffice?)

Secondly, and more importantly, there is no order on $\mathsf{F}$ unlike how there is an order on $\mathbb{R}$. That is, it doesn't make sense to talk about some function in $\mathsf{F}$ being "lesser than" (or "prior to") some other function in the set $\mathsf{F}$. Conversely, it is meaningful to say that some real number is lesser than another.

Without such an order on its sets, from what I understand, we can't talk about differentiating or integrating $L$.

As a sidenote, I am learning this proof from John Taylor's Classical Mechanics. I'm having trouble as its description of a function is a little off. It talks about $f(x)$ being a function of $x$ if it `depends' on $x$, which is pretty vague. A function is really, a relation between elements in one set, with elements in another, and it's unclear what the sets are in this case, for the function $L$.

mouldyfart
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  • Are you happy with differentiating in two dimensions? There's no canonical ordering there either. – Chappers Apr 09 '17 at 22:33
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    I think you might be confusing two $L$s. One $L$ is the length of a curve. One domain for definition would be $C^1[0,1]$ and then we have $L : C^[0,1] \to [0,1]$ defined as you have above. Another $L$ is the one that appears in most discussions of the Euler Lagrange equations. In this case, one has some function $L: \mathbb{R} \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, and the goal is minimise something like $J(f) = \int_0^1 L(f(x),f'(x), x) dx$ over some space of functions such as $C^1[0,1]$. In the length of the curve, the relevant $L$ would be $L(a,b,c) = \sqrt{1+b^2}$. – copper.hat Apr 09 '17 at 23:19
  • As an aside, I have found that many discussions of the Euler Lagrange equations do not address numerous technical details. – copper.hat Apr 09 '17 at 23:44
  • Echoing user @copper.hat's above comment, OP seems to conflate the length functional $L$ and the Lagrangian function $L(f, f^{\prime}, x)$. – Qmechanic Apr 16 '17 at 19:37

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