My question is a general one concerning mathematical modeling and the calculation of the arc lenght. There is a function $ f\colon [a, b]\to\mathbb {R} $, where $ f (x) $ includes a lot of parameters.
I nondimenzionalize it by $ g:=\frac {f}{\bar {f}} $ and $ y =\frac {x}{\bar {x}} $, where $\bar {f} $ and $\bar {x} $ consist of parameters (in general $\bar {f} \neq \bar {x} $).
I calculate the arc lenght for $ g (y):=\frac {f}{\bar {f}}(\bar {x} y)$ over $[\frac {a}{\bar {x}},\frac {b}{\bar {x}}] $, which is
$$arcl := \int_{\frac {a}{\bar {x}}}^{\frac {b}{\bar {x}}}\sqrt {1+g'(y)}. $$
I now get a result for $ arcl $, but I wonder, how to redimensionalize this thing. How can I redimensionalize $ arcl $?
Maybe this topic is also interesting for someone out there. So I will scetch the solution shortly.
For $\bar{f}=\bar{x}$ it is really easy. The redimensionalization is just the multiplication with $\bar{f}=\bar{x}$.
But if I choose $\bar{f}\neq\bar{x}$, s.t. $g=\frac{f}{\bar{f}}$ and $y=\frac{x}{\bar{x}}$, then we have to consider how we get the arc lenght.
We can calculate the arc length of a function by approximating it with line segments from $g(y_i)$ to $g(y_{i+1})$, where the union of all $[y_i,y_{i+1}]$ is just the interval $[\frac{a}{\bar{x}},\frac{b}{\bar{x}}]$ and $\frac{a}{\bar{x}}=y_0<...<y_n=\frac{b}{\bar{x}}$. These line segments are of the lenght
$$\sqrt{(g(y_{i+1})-g(y_i))^2+(y_{i+1}-y_i)^2}.$$
We have to form a Riemann-sum to get the integral. Here is an approach: https://en.wikipedia.org/wiki/Arc_length
If we follow this approach, we can easily see, how to redimensionalize the arc length of the function.
Please recognize that this is only a scetch of the solution, which might give an idea how to solve the problem.