Abstract
Hamiltonian systems with (strictly) more than two degrees of freedom do not admit a confinement of the phase space by invariant tori. For example, for a three-dimensional Hamiltonian system, the phase space has dimension 6, the constant energy level is five-dimensional and therefore the three-dimensional KAM tori do not separate the phase space by invariant regions. As a consequence the motion can diffuse through the invariant tori and can reach arbitrarily far regions of the phase space. This phenomenon is known as Arnold’s diffusion (Section 8.1) and a theorem due to N.N. Nekhoroshev allows us to state that it takes place at least on exponentially long times (Section 8.2, see also [75]).
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