I am learning about the time translation invariance of the Hamiltonian. I read that
the time translation invariance is already manifest in the fact that our Hamiltonian is chosen an instantaneous function of time—we have assumed that the dynamics only depends on position and velocity at the current time, rather than the full history of the particle’s trajectory.
I do not quite understand this. The Hamiltonian for a one-dimensional point particle can be written as $$H(x,t)=\frac{[p(x,t)]^2}{2m}+V(x,t)$$ where $p(x,t)$ and $V(x,t)$ are the momentum and potential repectively. If the time translation is $t \rightarrow t+ t_0$, the Hamiltonian will become $$H(x,t+t_0)=\frac{[p(x,t+t_0)]^2}{2m}+V(x,t+t_0).$$ Why can one say that $H(x,t)=H(x,t+t_0)$?