This booklet is designed to reveal from a normal and common perspective a spread ofmethods and effects pertaining to integrable platforms ofclassical me chanics. by way of such platforms we suggest Hamiltonian platforms with a finite variety of levels of freedom owning sufficiently many conserved amounts (in tegrals ofmotion) in order that in precept integration ofthe correspondingequa tions of movement should be decreased to quadratures, i.e. to comparing integrals of identified services. The research of those platforms was once a huge line ofstudy within the final century which, between different issues, encouraged the looks of the idea ofLie teams. Early in our century, in spite of the fact that, the paintings ofH. Poincare made it transparent that international integrals of movement for Hamiltonian platforms exist in basic terms in unparalleled situations, and the curiosity in integrable structures declined. until eventually lately, just a small quantity ofsuch structures with or extra de grees of freedom have been recognized. within the final fifteen years, notwithstanding, extraordinary development has been made during this path as a result invention by way of Gardner, Greene, Kruskal, and Miura [GGKM 19671 ofa new method of the integra tion ofnonlinear evolution equations referred to as the inverse scattering strategy or the strategy of isospectral deformations. utilized to difficulties of mechanics this system published the entire in tegrability of various classical platforms. it may be mentioned that every one structures of this type came across to this point are on the topic of Lie algebras, even though frequently this dating isn't really sosimpleas the oneexpressed through the well known theorem of E. Noether.
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