By V.I. Arnol'd, S.P. Novikov, A.G. Reyman, M.A. Semenov-Tian-Shansky, A.T. Fomenko, V.Ya. Gershkovich, M.A. Olshanetsky, A.M. Perelomov, A.G. Rejman, V.V. Trofimov, A.M. Vershik

This quantity comprises 5 surveys on dynamical platforms. the 1st one bargains with nonholonomic mechanics and offers anupdated and systematic remedy ofthe geometry ofdistributions and of variational issues of nonintegrableconstraints. the fashionable language of differential geometryused through the survey permits a transparent and unifiedexposition of the sooner paintings on nonholonomic problems.There is an in depth dialogue of the dynamical propertiesof the nonholonomic geodesic circulate and of assorted relatedconcepts, similar to nonholonomic exponential mapping,nonholonomic sphere, etc.Other surveys deal with numerous facets of integrableHamiltonian platforms, with an emphasis on Lie-algebraicconstructions. one of the themes coated are: the generalizedCalogero-Moser structures according to root platforms of easy Liealgebras, a ge- neral r-matrix scheme for constructingintegrable platforms and Lax pairs, hyperlinks with finite-gapintegration concept, topologicalaspects of integrablesystems, integrable tops, and so on. one of many surveys offers athorough research of a kin of quantum integrable systems(Toda lattices) utilizing the equipment of representationtheory.Readers will locate the entire new differential geometric andLie-algebraic tools that are at the moment utilized in the theoryof integrable platforms during this booklet. will probably be indispensableto graduate scholars and researchers in arithmetic andtheoretical physics.

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