Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds

Classical and Quantum Aspects

  • Anatoliy K. Prykarpatsky
  • Ihor V. Mykytiuk

Part of the Mathematics and Its Applications book series (MAIA, volume 443)

Table of contents

  1. Front Matter
    Pages 1-16
  2. Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk
    Pages 17-60
  3. Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk
    Pages 61-159
  4. Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk
    Pages 161-251
  5. Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk
    Pages 253-301
  6. Back Matter
    Pages 555-559

About this book

Introduction

In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi­ Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).

Keywords

Algebra Lie-Algebra differential equation differential geometry dynamical systems dynamics dynamische Systeme geometry manifold mathematical physics mechanics ordinary differential equation topological group

Authors and affiliations

  • Anatoliy K. Prykarpatsky
    • 1
    • 2
  • Ihor V. Mykytiuk
    • 2
    • 3
  1. 1.Institute of MathematicsUniversity of Mining and MetallurgyCracowPoland
  2. 2.Institute for Applied Problems of Mechanics and Mathematics of the NASLvivUkraine
  3. 3.Lviv Polytechnic State UniversityLvivUkraine

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-011-4994-5
  • Copyright Information Kluwer Academic Publishers 1998
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-010-6096-7
  • Online ISBN 978-94-011-4994-5
  • About this book
Industry Sectors
Electronics
Aerospace
Oil, Gas & Geosciences