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Mathematics and Its Applications

Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds

Classical and Quantum Aspects

Authors: Prykarpatsky, A.K., Mykytiuk, I.V.

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  • ISBN 978-94-011-4994-5
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Hardcover $159.99
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  • ISBN 978-0-7923-5090-3
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Softcover $149.00
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About this book

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).

Table of contents (5 chapters)

Table of contents (5 chapters)
  • Dynamical systems with homogeneous configuration spaces

    Pages 17-60

    Prykarpatsky, Anatoliy K. (et al.)

  • Geometric quantization and integrable dynamical systems

    Pages 61-159

    Prykarpatsky, Anatoliy K. (et al.)

  • Structures on manifolds and algebraic integrability of dynamical systems

    Pages 161-251

    Prykarpatsky, Anatoliy K. (et al.)

  • Algebraic methods of quantum statistical mechanics and their applications

    Pages 253-301

    Prykarpatsky, Anatoliy K. (et al.)

  • Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds

    Pages 303-553

    Prykarpatsky, Anatoliy K. (et al.)

Buy this book

eBook $119.00
price for USA in USD
  • ISBN 978-94-011-4994-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.99
price for USA in USD
  • ISBN 978-0-7923-5090-3
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
Softcover $149.00
price for USA in USD
  • ISBN 978-94-010-6096-7
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds
Book Subtitle
Classical and Quantum Aspects
Authors
Series Title
Mathematics and Its Applications
Series Volume
443
Copyright
1998
Publisher
Springer Netherlands
Copyright Holder
Springer Science+Business Media Dordrecht
eBook ISBN
978-94-011-4994-5
DOI
10.1007/978-94-011-4994-5
Hardcover ISBN
978-0-7923-5090-3
Softcover ISBN
978-94-010-6096-7
Edition Number
1
Number of Pages
559
Topics