Hamiltonian dynamics (Q2713571)

This book discusses hamiltonian dynamics while keeping a special focus on integrable systems, and aims to present a modern geometric and algebraic approach to these.NEWLINENEWLINENEWLINEPart I discusses Analytical Mechanics in about 120 pages. The treatment goes first -- in chapters 1 to 3 -- through basic and standard aspects of Newtonian, Lagrangian and Hamiltonian dynamics, with special discussions of constraints, of calculus of variations, of equivalent lagrangians, of direct and inverse Legendre transformations, and of the Jacobi-Poisson structure. After this, transformation theory for hamiltonian dynamics (that is, canonical and symplectic transformations) is discussed in chapter 4; chapter 5 is devoted to integration methods, in particular Hamilton-Jacobi method and Arnold-Liouville theorem.NEWLINENEWLINENEWLINEPart II discusses basic ideas of Differential Geometry in about 100 pages. It starts by introducing manifolds and tangent spaces in chapter 5; this includes submanifolds, maps between manifolds, vector fields, fiber bundles, Lie derivatives, and the Frobenius theorem. Chapter 6 is devoted to differential forms and to the metric structure of manifolds. In chapter 7 the reader is led to consider integration theory, and chapter 8 introduces Lie groups and algebras.NEWLINENEWLINENEWLINEPart III is called ``Geometry and Physics'' and is about 80 pages long. It starts, in chapter 9, by discussing symplectic manifolds and hamiltonian systems, including another discussion of the Arnold-Liouville theorem and a characterization of completely integrable systems in terms of invariant mixed tensor fields; it also introduces recursion operators and Nijenhuis structures as tools to discuss complete integrability, with applications to Kepler and rigid body dynamics, and gives the Magri theorems on Poisson-Nijenhuis structures. Chapter 10 is devoted to the orbit method, with application to the rigid body. The final chapter of this section briefly discusses classical electrodynamics.NEWLINENEWLINENEWLINEPart IV is devoted to Integrable Field Theories and lasts about 100 pages. It starts by thoroughly discussing the KdV equation, its symmetries and conservation laws, the Lax approach, inverse scattering method, and its hamiltonian structure. After setting safe grounds in chapter 12 on this special example, chapter 13 discusses general structures for the integrability problem, and chapter 14 gives a discussion on recursion operators. Chapter 15 discusses different topics, such as tensorial versions of the Lax representation, Liouville integrability for the Schrödinger equation, deformations of Lie algebra and integrable systems on coadjoint orbits. The final chapter 16 discusses integrability of fermionic dynamics.NEWLINENEWLINENEWLINEThe book is completed by some appendices, by short biographic notes of all the non-contemporary mathematicians and physicists whose names are mentioned, and by a bibliography comprising 60 books and about 150 papers.NEWLINENEWLINENEWLINESome examples, such as rigid body and Kepler dynamics, are repeatedly looked at from a variety of points of view; this will certainly help the student to grasp the new methods and concepts being introduced along the text. These include some quite recent developement, and others which are by now standard in the research literature but are still hard to find in textbooks.NEWLINENEWLINENEWLINEIn the whole, the author focuses on a specific issue of hamiltonian dynamics, namely complete integrability; it appears that here the geometric and algebraic (as opposed to analytical) approach is in the forefront. On the other hand, as the author also recalls, very little or nothing is said about nonintegrable systems and the modern qualitative theory of hamiltonian systems.