Arc length paramatrizations satisfy original system of differential equations?
Say we have a system of differential equations
on an interval , along with the restriction that
Now, Euler proved that as long as are continuous and not both zero at a point, then we can re-parametrize to some functions which have the same image as but are arc-length parametrized.
Now, say we have a solution to
that satisfies the hypotheses needed to apply the previously mentioned re-parametrization. Say are this re-parametrization, so they have the same image of on and satisfy
Is it still the case that