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• W2080363037 abstract "We present an outline of our recent work on Large Poincaré Systems (LPS) which form an important class of “non-integrable” systems as classified by Poincaré (1889). They are characterized by a continuous spectrum and the presence of continuous sets of resonances. As is well known, non-integrability leads to the appearance of random trajectories and chaos. Non-integrability is due to resonances responsible for divergences (the problem of “small denominators”). Generally, resonances are associated with a coupling parameter λ; then the KAM theory shows that for sufficiently small values of λ “most” trajectories remain periodic. On the contrary for LPS, almost all trajectories are random. We show that in this case the problem of the small denominators can be solved and new methods for the integration of these systems can be developed. These methods take into account the chaotic motion of the various degrees of freedom involved. These methods are applicable both to classical and quantum systems. They permit the solution of the eigenvalue problem associated with the time evolution of the distribution functions (the Liouville-von Neumann equation) in a constructive fashion, i.e., through a perturbation expansion analytic in the coupling constant. This can however be only done using non- unitary transformations (as the Poincaré's theorem shows that unitary transformations lead to divergences). The specific form of the non-unitary transformation is obtained by a suitable time ordering of the dynamical states. Integration in the Newton-Schrödinger sense is based on the use of time as a parameter for the motion of the dynamical object (or the wave function associated with it). Here we need in addition a second time related to the flow of correlations involving an ever increasing number of degrees of freedom. We may say that, in addition to the Newton-Schrödinger time, we have to introduce a second “internal” time describing the relations (the “correlations”) between the particles. Our dynamical description refers to a statistical description. This situation is similar to that of the radioactivity, where the behavior of individual atoms cannot be predicted. This is quite consistent as the radioactive decay or spontaneous emission correspond indeed to LPS. We now find this type of behavior both in quantum and classical theory as the result of resonances and chaos on the microscopic level." @default.
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• W2080363037 date "1991-01-01" @default.
• W2080363037 modified "2023-09-23" @default.
• W2080363037 title "Integrability and chaos in classical and quantum mechanics" @default.
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• W2080363037 doi "https://doi.org/10.1016/0960-0779(91)90052-b" @default.
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