# Is Newton's laws formulated using laboratory time?

The second Newton's law can be written as (in SI units) $$\frac{d}{dt}\vec p = \vec F.$$ Newton was considered Galilean transformations and the existence of a "absolute" time. Now suppose that this formulation must be independent (or ignore) Galilean transformations (and any other type of transformations, including Lorentz transformation).

Then my question is: the parameter $$t$$ in the above expression is the time measured by the clock of the laboratory?

I imagine that the some question can be applied to the expression for the velocity $$\frac{d}{dt}\vec r = \vec v.$$

I also imagine that the answer is "no", since for a onto function $$t\mapsto\tau = s(t)$$ the time derivative change as $$\frac{d}{dt} = \left(\frac{d}{d\tau}s^{-1}(\tau)\right)\frac{d}{d\tau}$$ and therefore two observers can discord about the magnitude of the force/velocity. The parameter is therefore the proper time define such that $$dt$$ is proportional to line element of the trajectory since this quantity is invariant under reparametrization of the trajectory in any riemannian manifold.