In the following question
We have in the answer, an arclength parametrization on a particular Catenoid that is given by : $(u'(t))^2+cosh^2(u(t))(v'(t))^2=1$
(Considering we parametrize the catenoid by $r(u,v)=(cosh(u)cos(v),cosh(u)sin(v),u)$)
My question is what principle or formula is used to determine such an arc lenght curve expression ,
To be more precise , what i want is to find expression of the "curve that generates the surface of revolution when turning it around an axis ", and i guess it has to be arc lenght parametrized to be able to apply Eulers equation and minimize the related integral ,
Look at your parameterization $r(u,v)$. If you think about how those coordinates describe the geometry of the catenoid, taking $v=\text{constant}$ will give you a curve that will generate a catenoid as a surface of revolution. Taking $v=0$, it is easy to see that the function $x(z) = \cosh z$ will suffice.
If you have some curve $\gamma(t)$ in a Riemannian space with metric $g$, you can find the arclength $s$ of the curve at any time $T$ as
$$s(T) = \int_{t_0}^{T}\|\vec{v}\|dt$$ where $\vec{v} = \gamma '(t)$. This becomes clear using the notion that distance traveled equals the integral of the speed with respect to time. Using a tensor formulation, we will say that the length of the velocity vector is $$\|\vec{v}\| = \sqrt{g_{ij} v^i v^j }$$ thus
$$s(T) = \int_{t_0}^{T} \sqrt{g_{ij} v^i v^j } dt \text{ .}$$
We can reparametrize $\gamma(t)$ as $\gamma(s) = \gamma(T(s))$ by integrating and inverting $s(T)$.
Use the geodesic equation as parameterized by arclength. The geodesic equation is a statement that the second derivative of the position with respect to arclength equals zero. It reads as
$$ \frac{d^2 x^k}{ds^2} + \Gamma^k_{ij} \frac{dx^i}{ds} \frac{dx^j}{ds} = 0 $$
which is (in general) a system of coupled, second order, ordinary differential equations. Here $\Gamma^k_{ij}$ is the Christoffel symbol which are the connection coefficients for this geometry. Solutions to the geodesic equation will, by nature, be parameterized with respect to arclength.