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Lecture Notes of the Unione Matematica Italiana

Elementary Symplectic Topology and Mechanics

Authors: Cardin, Franco

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  • ISBN 978-3-319-11026-4
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About this book

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.

Reviews

“This book brings the reader from the basics of symplectic topology through several concepts and issues relative to Hamiltonian and Lagrangian systems, Hamilton-Jacobi equations, the calculus of variations, and Morse and Lusternik-Schnirelman theories. … The book seems best suited to researchers and students from graduate level onwards.” (Maria Letizia Bertotti, Mathematical Reviews, August, 2015)


Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook 32,12 €
price for Spain (gross)
  • ISBN 978-3-319-11026-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 41,59 €
price for Spain (gross)
  • ISBN 978-3-319-11025-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Elementary Symplectic Topology and Mechanics
Authors
Series Title
Lecture Notes of the Unione Matematica Italiana
Series Volume
16
Copyright
2015
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-11026-4
DOI
10.1007/978-3-319-11026-4
Softcover ISBN
978-3-319-11025-7
Series ISSN
1862-9113
Edition Number
1
Number of Pages
XVII, 222
Number of Illustrations
33 b/w illustrations, 11 illustrations in colour
Topics