Book Review
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Pure and Applied Geophysics
Book Review Nonlinear Dynamical Systems of Mathematical Physics by Denis Blackmore, Anatoliy K. Prykarpatsky and Valeriy Hr Samolyenko, Spectral and Symplectic Integrability Analysis, World Scientific, 2011; ISBN: 13 978-981-4327-15-2 ANDRZEJ ICHA1 Physics concerns itself with the most fundamental principles and laws of the nature. But what is mathematical physics? The concise answer may be following: mathematical physics deals with the formulation and analysis of physical problems by means of mathematical constructions and apparatus. These include the application of existing mathematical techniques to physics problems and the development of new mathematical ideas for their study. Nonlinear Dynamical Systems of Mathematical Physics by D. Blackmore et al. is an excellent text devoted to nonlinear dynamical systems in context of the spectral and symplectic integrability analysis, addressed to advanced undergraduate and graduate students of exact and natural sciences. The book is organized into fourteen chapters. Chapter 1, General Properties of Nonlinear Dynamical Systems, is the terse introduction to the theory of dynamical systems on finite-dimensional symplectic manifolds. Basic mathematical concepts are presented and described concisely: the notion of the phase space of the dynamical system; the Liouville condition; the famous Poincare´ theorem; the Birkhoff–Khinchin theorem on ergodicity and an analogous theorem for a discrete dynamical systems; the Poissonian formalism; the Hamilton–Jacobi technique and the procedure of Dirac reduction of Poisson operators on submanifolds.
1 Pomeranian Academy in Słupsk, Institute of Mathematics, ul. Arciszewskiego 22d, Słupsk 76-200, Poland. E-mail: [email protected]
Chapter 2 focuses on the Geometric and Algebraic Properties of Nonlinear Dynamical Systems with Symmetry. Some remarks on the Poisson structures and Lie group actions on Poisson manifolds together with the research programme relating to dynamical systems possessing an intrinsic symmetry structure, are presented first. Next, the canonical reduction method applied to geometric forms on symplectic manifolds with symmetry and associated canonical connections are discussed. The special fiber-bundle structure used to the study of the motion of a charged particle under an abelian Yang–Mills type gauge field equations, is analysed also. This approach is known as the principle of minimal interaction and is very useful for studying various interacting physical systems. The last section introduces the reader to the world of classical and quantum integrable systems. Starting with the technical definition of an integrable classical system provided by the well-known Liouville theorem, the basic aspects of integrability are studied including elements of the Hopf and quantum algebras theory. On the basis of the structure of Casimir elements associated with Hopf algebras, an integrable flows connected with the naturally induced Poisson structures on an arbitrary co-algebra and their deformations, a
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