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A memoir on integrable systems. (English) Zbl 1417.37013
Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-3-540-59000-2/hbk; 978-3-540-68912-6/ebook). 280 p. (2017).
Publisher’s description: Integrable dynamical systems are usually associated with Hamiltonian ones. The present book considers the bigger class of systems which are not (at least a priori) Hamiltonian but possess tensor invariants, in particular, an invariant measure. Such systems are as rare as Hamiltonian ones that have additional first integrals and therefore must be considered as number one candidates for integrable problems. Several integrability theorems related to the existence of tensor invariants are formulated. The authors display the geometrical background of some classical and new hierarchies of integrable systems and give their explicit solution in terms of theta-functions. Most of the results discussed in this book have not been published before, so that this book will be immensely useful both to specialists in analytical dynamics who are interested in integrable problems and those in algebraic geometry who are looking for applications.

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37Jxx Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70Hxx Hamiltonian and Lagrangian mechanics