Adiabatic invariants and their preservation under symplectic discretizations

We will apply normal form theory to highly oscillatory Hamiltonian systems and discuss the preservation of adiabatic invariants over exponentially long periods of time. Next, the conservation of adiabatic invariants under a symplectic discretization will be considered. Backward error analysis will provide us the appropriate tool to show that, again, the invariants are preserved over exponentially long periods of time.