Given a general curve $\gamma$, we know that if it is parametrized by arclength then $||\gamma'(t)||=1$, for all $t$. My question now is: given a curve $\gamma$, if $||\gamma'(t)||=1$, can we conclude that $\gamma$ is parametrized by arclength?
If not, how can we find out, given a general curve, if it is parametrized or not by arclength?
Yes, a curve is parametrized by arclength if and only if $\| \dot{\gamma}(t)\|=1$ for all $t$. To see why, recall that the length of the curve from the initial time $t_0$ to the current time $t$ is given by the formula $$ \verb+length +(t)=\int_{t_0}^t \|\dot{\gamma}(\tau)\|\, d\tau , $$ and arclength parameterization means exactly that $\verb+length +(t) = t-t_0$, which happens precisely when the integrand function is identically $1$.