Abstract
It is shown that two-degree-of-freedom Hamiltonian systems of the billiard type are equivalent to adiabatically varying one-degree-of-freedom Hamiltonian systems for solutions staying near the boundary. Under some nondegeneracy conditions such systems possess a large set of quasiperiodic solutions filling out two-dimensional invariant tori. The latter separate the extended phase space into layers providing stability for all time. The result is illustrated on a few examples.
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