Elementary Symplectic Topology and Mechanics by Franco CardinElementary Symplectic Topology and Mechanics by Franco Cardin

Elementary Symplectic Topology and Mechanics

byFranco Cardin

Paperback | December 12, 2014

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This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.
Title:Elementary Symplectic Topology and MechanicsFormat:PaperbackDimensions:222 pagesPublished:December 12, 2014Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:331911025X

ISBN - 13:9783319110257

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Table of Contents

Beginning.- Notes on Differential Geometry.- Symplectic Manifolds.- Poisson brackets environment.- Cauchy Problem for H-J equations.- Calculus of Variations and Conjugate Points.- Asymptotic Theory of Oscillating Integrals.- Lusternik-Schnirelman and Morse.- Finite Exact Reductions.- Other instances.- Bibliography.