








1. 
Record Nr. 
UNINA9910299778503321 


Autore 
Cardin Franco 


Titolo 
Elementary Symplectic Topology and Mechanics [[electronic resource] /] / by Franco Cardin 







Pubbl/distr/stampa 


Cham : , : Springer International Publishing : , : Imprint : Springer, , 2015 









ISBN 






Edizione 
[1st ed. 2015.] 





Descrizione fisica 

1 online resource (237 p.) 






Collana 

Lecture Notes of the Unione Matematica Italiana, , 18629113 ; ; 16 






Disciplina 








Soggetti 

Mathematical physics 
Differential geometry 
Calculus of variations 
Mathematical Physics 
Differential Geometry 
Calculus of Variations and Optimal Control; Optimization 








Lingua di pubblicazione 






Formato 
Materiale a stampa 





Livello bibliografico 
Monografia 





Note generali 

Description based upon print version of record. 






Nota di bibliografia 

Includes bibliographical references. 






Nota di contenuto 

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








Sommario/riassunto 

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 MaslovHörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of HamiltonJacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds 










belonging to distinct welldefined 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. 





 