On phase at a resonance in slow-fast Hamiltonian systems
Yuyang Gao (Loughborough)
We consider a slow-fast Hamiltonian system with one fast angular variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Trajectories of the system averaged over the fast phase cross the resonant surface. The fast phase makes ∼ 1 ε turns before arrival at the resonant surface (ε is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival in the resonance was previously derived in the study of charged particle dynamics based on heuristic considerations without accuracy estimates. We provide a rigorous derivation of this formula and √ prove that its accuracy is O( ε), up to a logarithmic correction. This estimate for the accuracy is optimal.
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