Preservation of adiabatic invariants under symplectic discretization

Symplectic methods, like Verlet's method, are standard tools for long time integration of Hamiltonian systems arising, for example, in molecular dynam ics. 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 sc ales that is also preserved under symplectic discretization. Specifically, highly oscillatory degrees of freedom often lead to almost preserved quanti ties (adiabatic invariants). Using recent results from backward error analy sis and normal form theory, we show that a symplectic method preserves thos e adiabatic invariants. We also discuss step size restrictions necessary to maintain adiabatic invariants in practice. (C) 1999 Elsevier Science B.V. and IMACS. All rights reserved.