Before revision:  Finally, we show that the majority of typical discretizations of this ODE , such as the Heavy ball method, do not possess suitable discrete Lyapunov functions , and hence fail to reproduce the desired limiting behaviour of this ODE, which in turn implies that their converge rates when seen as optimization methods cannot behave in an "accerelated" manner .
After revision:  Finally, we show that the majority of typical discretizations of this ODE , such as the Heavy ball method, do not possess Lyapunov functions with properties similar to those of the Lyapunov function constructed here for the Nesterov method .