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Lectures on Symplectic Geometry

By Da Silva, Ana Cannas

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Book Id: WPLBN0000660343
Format Type: PDF eBook
File Size: 1.02 MB
Reproduction Date: 2005
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Title: Lectures on Symplectic Geometry  
Author: Da Silva, Ana Cannas
Volume:
Language: English
Subject: Science., Mathematics, Logic
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Da Silva, A. C. (n.d.). Lectures on Symplectic Geometry. Retrieved from http://www.newspaperslibrary.net/


Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: A symplectic form is a 2-form satisfying an algebraic condition ? nondegeneracy ? and an analytical condition ? closedness. In Lectures 1 and 2 we define symplectic forms, describe some of their basic properties, introduce the first examples, namely even-dimensional Euclidean spaces and cotangent bundles.

Table of Contents
CONTENTS vii Contents Foreword v Conventions vi Acknowledgments vi I Introduction 1 1 Symplectic Forms 1 1.1 Skew-Symmetric Bilinear Maps . . . . . . . . . . . . . . . . . . . 1 1.2 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Symplectomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 5 Homework 1: Symplectic Linear Algebra 6 2 Symplectic Form on the Cotangent Bundle 7 2.1 Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Tautological and Canonical Forms in Coordinates . . . . . . . . . 7 2.3 Coordinate-Free Definitions . . . . . . . . . . . . . . . . . . . . . 8 2.4 Naturality of the Tautological and Canonical Forms . . . . . . . 9 Homework 2: Symplectic Volume 11 II Symplectomorphisms 13 3 Lagrangian Submanifolds 13 3.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Lagrangian Submanifolds of T X . . . . . . . . . . . . . . . . . . 14 3.3 Conormal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Application to Symplectomorphisms . . . . . . . . . . . . . . . . 17 Homework 3: Tautological Form and Symplectomorphisms 18 4 Generating Functions 20 4.1 Constructing Symplectomorphisms . . . . . . . . . . . . . . . . . 20 4.2 Method of Generating Functions . . . . . . . . . . . . . . . . . . 21 4.3 Application to Geodesic Flow . . . . . . . . . . . . . . . . . . . . 22 Homework 4: Geodesic Flow 25

 

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