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Where does this identity come from (natural parametrization)

Mathematics Asked by Feynman_00 on January 5, 2021

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

Natural parametrization

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

come from?

One 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

$$s=int_{u_1}^{u_2}|dot z|du$$

Answered by Cye Waldman on January 5, 2021

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