Dynamical Systems VII: Integrable Systems, Nonholonomic Dynamical Systems

V.I. Arnol’d, S.P. Novikov, A.G. Reyman, M.A. Semenov-Tian-Shansky, 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. Vershik3540181768, 9783540181767

This volume contains five surveys on dynamical systems. The first one deals with nonholonomic mechanics and gives anupdated and systematic treatment ofthe geometry ofdistributions and of variational problems with nonintegrableconstraints. The modern language of differential geometryused throughout the survey allows for a clear and unifiedexposition of the earlier work on nonholonomic problems.There is a detailed discussion of the dynamical propertiesof the nonholonomic geodesic flow and of various relatedconcepts, such as nonholonomic exponential mapping,nonholonomic sphere, etc.Other surveys treat various aspects of integrableHamiltonian systems, with an emphasis on Lie-algebraicconstructions. Among the topics covered are: the generalizedCalogero-Moser systems based on root systems of simple Liealgebras, a ge- neral r-matrix scheme for constructingintegrable systems and Lax pairs, links with finite-gapintegration theory, topologicalaspects of integrablesystems, integrable tops, etc. One of the surveys gives athorough analysis of a family of quantum integrable systems(Toda lattices) using the machinery of representationtheory.Readers will find all the new differential geometric andLie-algebraic methods which are currently used in the theoryof integrable systems in this book. It will be indispensableto graduate students and researchers in mathematics andtheoretical physics.