Version of 21 July 2015 Long-term analysis of the St¨ormer–Verlet method for Hamiltonian systems with a solution-dependent high frequency Ernst Hairer · Christian Lubich Abstract The long-time behaviour of the St¨ormer–Verlet–leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved: The considered Hamil- tonian systems have the action as an adiabatic invariant over long times that cover arbitrary negative powers of the small parameter. The St¨ormer–Verlet method approximately conserves a modified action and a modified total en- ergy over a long time interval that covers a negative integer power of the small parameter. This power depends on the size of the product of the stepsize with the high frequency. Keywords Oscillatory Hamiltonian system · Modulated Fourier expansion · St¨ormer-Verlet scheme · Leapfrog integrator · Long-time energy conservation · Adiabatic invariant Mathematics Subject Classification (2000) 65P10 · 65L05 · 34E13 1 Introduction In the last decade much insight into the properties of numerical integrators for highly oscillatory Hamiltonian systems has been gained, starting by thoroughly This work has been supported by the Fonds National Suisse, Project No. 200020-144313/1. Ernst Hairer Section de math´ematiques, 2-4 rue du Li`evre, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland. E-mail: Ernst.Hairer@unige.ch Christian Lubich Mathematisches Institut, Universit¨at T¨ ubingen, Auf der Morgenstelle, D-72076 T¨ ubingen, Germany. E-mail: Lubich@na.uni-tuebingen.de