Arc-length integral with curve giving extremal value

Leb

1. The problem statement, all variables and given/known data
View attachment B3.bmp
For whoever does not want to read the attached problem:

Firstly, I need to express the arc-length from given $x=r\cos\theta, y=r\sin\theta z = f(r), \text{ where } f(r) \text{ is an infinitely differentiable function and } r=r(\theta) \text{ i.e. parameter is } \theta$ I begin with expressing it as $ds = \sqrt{ r^{2} + r'^{2} + f'^{2} }$ is that correct ?

I then need to show that the solutions of those curves are given by:
$r ' ^{2} = r^{2}\frac{\left( \frac{r^{2}}{C} - 1\right)}{1 + f ' ^{2)}}$

2. Relevant equations

Lagrangian is independent of theta in this case, so I think I can use
$\frac{d}{d\theta}\left( L - r' \frac{\partial{L}}{\partial{r'}} \right) = 0$

3. The attempt at a solution

Using the above, the term in parenthesis is equal to a constant. Differentiating L w.r.t. r' and rearranging I get $r'^{2} = \frac{r^{4}+2f'^{2}r^{2} + f'^{4}-Br^{2} - Bf'{2}}{B}$ where B is a constant. I thought that this uses the good old multiply by "1" trick, but when I tried it it got no cancellations.

I can see the $r^{2}\left( \frac{r^{2}}{C} - 1\right)$ term in there, but do not know how to get the rest...

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voko

For the line element ds the following is true $ds^2 = dx^2 + dy^2 + dz^2$. Since your curve is on a surface parametrized by $r$ and $\theta$, and since $r$ itself is a function of $\theta$, you should end up with $ds = g(\theta)d\theta$.

Leb

Do you mean I should express $\frac{df}{d\theta} \text{ as } \frac{df}{dr}\frac{dr}{d\theta}$ ?

voko

What I mean is that the expression for ds you got is incorrect. You can derive the correct equation from the expression for ds in the Cartesian coordinates by using the chain rule.

voko

$$ds^2 = dx^2 + dy^2 + dz^2 = (d(r \cos \theta))^2 + (d(r \sin \theta))^2 + (d(f(r)))^2 = (dr \cos \theta - r \sin \theta d\theta)^2 + (dr \sin \theta + r \cos \theta d\theta)^2 + (f'dr)^2 \\ = dr^2 + r^2 d\theta^2 + f'^2 dr^2 = (1 + f'^2)dr^2 + r^2d\theta^2 \\ = (1 + f'^2)(dr(\theta))^2 + r^2d\theta^2 = (1 + f'^2)(r'd\theta)^2 + r^2d\theta^2 = ((1 + f'^2)r'^2 + r^2)d\theta^2 \\ ds = \sqrt {(1 + f'^2)r'^2 + r^2}d\theta$$

Leb

Thanks for taking the time to type that, but if we parameterize w.r.t. theta, why do we have d \theta ?

voko

Parametrized by $\theta$ means that the length is a function of $\theta$: $s = s(\theta)$, thus $ds = s'd\theta$.

Leb

So I take it that the previous link is rubbish ? Thanks again !

voko

No, it is not rubbish. If you look carefully, the formula I derived equals their formula if f = const. Your problem has three dimensions, not two as they have.

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voko

Why should it disappear? Take a very simple example: $y = x^2$. You can say that "y is parametrized by x". Then $dy = 2x dx$; x does not disappear. Even if $y = x$, you still have $dy = dx$.

Leb

Well, I am a numpty and always had particular trouble with notation of derivatives and using the "implicit" chain rule.

Anyway, thanks very much for bothering to reply !

Leb

Hehe, reparametrization... I always viewed the parameters as an "arbitrary" choice when we are talking about simple curvy curves (not circles or ellipses).

"Arc-length integral with curve giving extremal value"

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