Time reversal symmetry and energy drift in conservative systems

According to some authors, Time Reversal Symmetry (TRS) is a murky problem. By considering that usually the TRS, to which they refer to, regards continuous variation of time, it is easy to estrapolate that TRS in the discrete domain is even more murky. In Numerical Analysis we come across to this problem, usually confined in the Physics domain, when trying to approximate the solutions of Hamiltonian problems (or, more generally, conservative problems). Often, for such problems an energy drift is observed even when using the most accurate numerical methods. It was believed that Hamiltonian systems satisfyng TRS (in the form established by the currently used definition) would prevent the drift, but recent counterexamples have shown that this is not always the case. In this paper we intend to analyze the problem, starting from the most current definition of TRS.