Preservation of Adiabatic Invariants under Symplectic Discretization

Symplectic methods, like the Verlet method, are a standard tool for the long term integration of Hamiltonian systems as they arise, for example, in molecular dynamics. One of the reasons for the popularity of symplectic methods is the conservation of energy over very long periods of time up to small fluctuations that scale with the order of the method. In this paper, 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, like the Verlet method, preserves those adiabatic invariants. We also discuss step-size restrictions necessary to maintain adiabatic invariants in practical computations.