# Newton's equation under time translation

I'm struggling in Arnold's mathematical methods of classical mechanics when he's talking about the covariance of Newton's equation under galilean transformations (Newton's equation is $$\boldsymbol{\ddot{x}}=\boldsymbol{F}(\boldsymbol{x}, \boldsymbol{\dot{x}},t)$$). Here is an image of the relevant part

I don't see how it cannot depend on time. Can someone show another way of thinking to get to this conclusion. Does there exist a mathematical approach to get this?

The assumption is: If $$x(t)$$ is a solution of $$\ddot{x} = F(x,\dot x,t)$$ then $$x_s(t):= x(t+s)$$ is also a solution.
Now as Dale said, there is no a priori reason for this assumption to be true. It just makes sense physically. It is easy to show that in the case that the force $$F$$ can not explicitly depend on $$t$$ and I presume that this is the content of your question:
Let $$x$$ be a solution of the $$\ddot{x} = F(x,\dot x,t)$$. The chain rule implies $$\dot x_s(t) = \dot x(t+s),\ddot x_s(t) = \ddot x(t+s)$$. Because of the assumption we have $$\ddot x_s(t) = F(x_s(t),\dot x_s(t),t)$$ and this implies $$\ddot x(t+s) = F(x(t+s),\dot x(t+s), t) = F(x(t+s),\dot x(t+s),t+s)$$ Since $$t,s \in \mathbb{R}$$ were arbitrary this means that $$F$$ does not explicitely depend on $$t$$, only implicitly through its dependence on $$x(t),\dot x(t)$$. Thus we have $$\frac{\partial F}{\partial t} = 0$$ but $$\frac{dF}{dt} \neq 0$$ in general.