I remember doing side by side plots of conservative Hamiltonian trajectories doing a standard Euler method (maybe even RK45), vs a symplectic method (which will

I remember doing side by side plots of conservative Hamiltonian trajectories doing a standard Euler method (maybe even RK45), vs a symplectic method (which will maintains energy conservation). The RK45 implementation had a very nice symmetric pattern, but which was completely different from the one in the (correct) symplectic implementation. This was a useful eye opener for me to not just blindly rely on Matlab's ODE45 or other default solvers...