Preservation of adiabatic invariants under symplectic discretization Original Research Article

Symplectic methods, like Verletʹs method, are standard tools for long time integration of Hamiltonian systems arising, for example, in molecular dynamics. A reason for their popularity is conservation of energy over very long time up to small fluctuations that scale with the order of the method. We discuss a qualitative feature of Hamiltonian systems with separated time scales that is also preserved under symplectic discretization. Specifically, highly oscillatory degrees of freedom often lead to almost preserved quantities (adiabatic invariants). Using recent results from backward error analysis and normal form theory, we show that a symplectic method preserves those adiabatic invariants. We also discuss step size restrictions necessary to maintain adiabatic invariants in practice.