According to Wikipedia,
Two dynamical systems on the same state space, defined by $\dot {x}=f(x)$ and $\dot {x}=g(x)$, are said to be orbitally equivalent if there is a positive function, $\mu :X\to \mathbb {R}$, such that $g(x)=\mu (x)f(x)$. Orbitally equivalent system differ only in the time parametrization.
I wonder why do orbitally equivalent have same orbits. Intuitively, the two systems would have the same "direction of velocity" at any given point, but this is not a mathematical proof, right?
I can not found any relating information except this question where lhf implicitly uses this property.
If $x=\xi(t)$ is a solution of $dx/dt=f(x)$, let $T(s)$ be a solution of $dT/ds=\mu(\xi(T(s)))$. Then $t=T(s)$ is a time reparametrization that does the trick for that particular orbit, i.e., $x=\xi(T(s))$ solves $dx/ds=g(x)$: $$ \frac{dx}{ds} =\frac{d}{ds} \Bigl( \xi(T(s)) \Bigr) =\frac{d\xi}{dt}(T(s)) \cdot \frac{dT}{ds}(s) = f(\xi(T(s))) \cdot \mu(\xi(T(s))) =g(\xi(T(s))) =g(x) . $$