Perturbation Theory and the Study of Systems Close to Integrable

1 Perturbation Theory and the Study of Systems Close to Integrable 

in the space of all Hamiltonians there exist open domains that sometimes occupy almost the whole space and consist of Hamiltonians / of general position for which the corresponding Hamiltonian systems sgrad / are not integrable in the sense of Liouville (nor in any other reasonable sense). The described picture is 

Nevertheless, the available results negative character, i.e., those declaring nonintegrability of many concrete types of systems, make it possible to think of the formulated principle as a certain experimental observation. In other words, integrable cases fill up the set of measure zero in the space of all systems. 

FVom this it is already clear that the search for integrable systems is an exceedingly complicated problem because in the boundless variety of all possible Hamiltonians one should somehow guess or algorithmically reveal those rare cases where certain additional symmetries of the system provoke the appearance of a sufficient number of integrals. That is the reason why we have narrated about the system integration problems in the first part of the book. 

We will now demonstrate that a Hamiltonian taken by rule of thumb most often generates a nonintegrable system, in any case if the description of motion of a three-dimensional heavy rigid body is meant. 

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