Symplectic Integrator and its Applications

The first- and the second-order symplectic integrators for the one-dimensional harmonic oscillator are reconstructed on the basis of effective Liouville operators, which can be defined only within the convergence radius. The first-order one for the $q^4$-potential system breaks down for different time steps depending on the initial condition, which indicates that no conservation value exists for the system in the first- order symplectic integrator.