Elementary Symplectic Topology and Mechanics

Franco Cardin

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English

Springer International Publishing AG
12 December 2014

Calculus of variations; Differential & Riemannian geometry; Mathematical physics

Series: Lecture Notes of the Unione Matematica Italiana

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.

By: Franco Cardin
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2015 ed.
Volume: 16
Dimensions: Height: 235mm, Width: 155mm, Spine: 13mm
Weight: 3.752kg
ISBN: 9783319110257
ISBN 10: 331911025X
Series: Lecture Notes of the Unione Matematica Italiana
Pages: 222
Publication Date: 12 December 2014
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Reviews for Elementary Symplectic Topology and Mechanics

“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)