On the geometry of transport in phase space. I. Transport in k-degree-of- freedom Hamiltonian systems, \(2\leq k<\infty\)
The transport theory for 2-dimensional Hamiltonian systems is facilitated by the existence of (complete or partial) barriers (invariant tori, stable and unstable manifolds, cantorus) which permit to estimate the transport rate in the phase space. Using the perturbation argument, the author determines a large class of higher dimensional Hamiltonian systems with quite analogous property: there exist certain normally hyperbolic invariant subsets of codimension one in every level set of the Hamiltonian. Then the mechanism of the transport (in particular the lobe dynamics, the conservation rules, turnstiles) can be investigated in the same manner as in the case of two-degree of freedom. A thorough discussion of the Hamiltonian \[ H_{\epsilon}=v^ 2-\cos \phi +y^ 2_ 1+\omega^ 2_ 1x^ 2_ 1+y^ 2_ 2+\omega^ 2_ 2x^ 2_ 2+\epsilon \tilde H \] and several general remarks concerning the transport near resonances and the Arnold diffusion conclude the paper.
scientific article; zbMATH DE number 4157526
scientific article