equivalent function to minimize the length in curved space

Question

Consider we have a certain metric. For instance: \begin{equation} ds^2 = \frac{dr^2}{r^2-Ml^2} +r^2d\phi^2. \end{equation}

In order to find the shortest length between two angles $\phi=\alpha$ and $\phi=\beta$, we have to minimize this function, \begin{equation} S = \int_\alpha^\beta d\phi \quad \sqrt{\frac{1}{r^2-Ml^2}\left(\frac{dr}{d\phi}\right)^2+r^2}. \end{equation}

Now because of the square root, the function to minimize is a bit complicated. Is there an equivalent function to minimize that is less complicated?

Answer 1

By application of Euler Lagrange equation for the functional $ f(y,y') = f $ ( when no x terms occur explicitly), we have

$$ f - y' \frac{\partial f}{ \partial y'}= C_1 \tag1 $$

and for the square rooted functional

$$ \sqrt{f} - y' \frac{\partial \sqrt{f}}{ \partial y'}= C_2 $$

Or

$$ f - \frac{y'}{2} \frac{\partial f}{ \partial y'}= C_2\sqrt{f} \tag 2 $$

(1) and (2) are quite different and in general cases lead to different ODEs. I believe there is no workaround.

Check for simple cases

$$ f= y'/y \text{ and } f= y^{'2}/y^2$$

result respectively in different ODEs

$$ C_1=0 \text{ and } C_2 y^2 + y^{'2}=0 $$

In the present case we have after applying Euler Lagrange

$$\frac{r^2}{ \sqrt{\frac{1}{(r^2-Ml^2)}\left(\frac{dr}{d\phi}\right)^2+r^2}} = C_1$$

which can be squared and simplified easily. May be elliptic function solutions.

Answer 2

If you have a scalar function $f$ of the form $$f(x) = g(x)^2$$ for another function $g$ then it is clear that $$f'(x) = 2\:g(x)\:g'(x) \text{ .}$$

So long as $g(x)\neq 0$ then the critical points of $f$ coincide with the critical points of $g$. When minimizing the square root of a quantity, we can most always square it first without consequence.

Above, I only talked about minimizing scalar functions. We can extend this reasoning to functionals of the form

$$ S[y(x),y'(x);x] = \int_\alpha^\beta f(y,y',x) \text{ d}x$$

by using the Fréchet derivative which is equipped with a chain rule of it's own. Applying the chain rule we would see that, in much the same manner, minimizing your functional $S$ would be equivalent to minimizing $$ L[y(x),y'(x);x] = \int_\alpha^\beta \big[f(y,y',x)\big]^2 \text{ d}x \text{ .}$$

We even find that we get the same condition as before that $f(y,y',x)\neq 0$ which you needn't worry about for any standard, Riemannian metric $\text{d}s^2$.