Boris numerical scheme due to its long-time stability,accuracy and conservative properties has been widely applied in many studies of magnetized plasmas.Such algorithms conserve the phase space volume and hence provide accurate charge particle orbits.However,this algorithm does not conserve the energy in some special electromagnetic configurations,particularly for long simulation times.Here,we empirically analyze the energy behavior of Boris algorithm by applying it to a 2D autonomous Hamiltonian.The energy behavior of the Boris method is found to be strongly related to the integrability of our Hamiltonian system.We find that if the invariant tori is preserved under Boris discretization,the energy error can be bounded for an exponentially long time,otherwise the said error will show a linear growth.On the contrary,for a non-integrable Hamiltonian system,a random walk pattem has been observed in the energy error.