Where does this identity come from (natural parametrization)

Question

Reading Analytical Mechanics by Fasano and Marmi, I bumped into this identity with natural parametrization.

Where does the identity
begin{equation}
s=int_0^sleft|frac{dboldsymbol{x}}{dsigma}right|dsigma
end{equation}
come from?

Cye Waldman · Answer

In the complex plane we can say that a differential length $ds$ is given by
$$ds = |dz| =bigg| frac{dz}{du}bigg| du =|dot z|du$$
and thus the arc length between two parametric points, say $u_{1,2}$ is thus
$$s=int_{u_1}^{u_2}|dot z|du$$

Where does this identity come from (natural parametrization)

One Answer

Add your own answers!

Ask a Question