Abstract
The orbit of a linear Hamiltonian system with one degree of freedom is closed, divergent to infinity exponentially, or divergent as a free motion. This paper proves that every symplectic integrator reproduces a similar phase portrait in the former two cases when the step size is small enouth. This fact is considered based on a one-parameter family of conservatives admitted by a discrete and continuous systems. Furthermore, the existence of a conservative is studied for certain nonlinear systems.
