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.
Usually in physics symmetries are not explained. They are simply observed and accepted and then used as the most fundamental explanations of other things.
The fundamental importance of symmetries is because of Noether’s theorem which established the connection between symmetries and conservation laws. In the case of time translation symmetry the corresponding conservation law is the conservation of energy. So if the laws did depend on time then energy would not be conserved.
There is, to my knowledge, no a priori reason that the laws of physics must have any given symmetry. We simply observe that they do and write our formulas accordingly.
