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Variational calculus / Jean-Pierre Bourguignon.
Springer Nature - Springer Mathematics and Statistics eBooks 2022 English International Available online
View online- Format:
- Book
- Author/Creator:
- Bourguignon, J.-P., author.
- Series:
- Springer monographs in mathematics.
- Springer monographs in mathematics
- Language:
- English
- Subjects (All):
- Variational inequalities (Mathematics).
- Physical Description:
- 1 online resource (284 pages)
- Edition:
- 1st ed.
- Place of Publication:
- Cham, Switzerland : Springer, [2022]
- Summary:
- This book provides a comprehensive introduction to the Calculus of Variations and its use in modelling mechanics and physics problems.Presenting a geometric approach to the subject, it progressively guides the reader through this very active branch of mathematics, accompanying key statements with a huge variety of exercises, some of them solved.
- Contents:
- Intro
- Editor's Preface
- Preface
- Preface to the English Edition
- To the Reader
- Contents
- Part I THE ANALYTIC SETTING
- Chapter I A First Generalisation of the Notion of Space: Spaces of Infinite Dimension
- A. A First Encounter with Infinite-Dimensional Vector Spaces
- B. A Useful Special Case: Normed Spaces
- C. Compact Spaces
- D. Looking For Compact Sets
- E. Historical Notes
- Chapter II Banach Spaces and Hilbert Spaces
- A. Cauchy Sequences and Complete Metric Spaces
- B. A Fundamental Category: Banach Spaces
- C. Dual Space and Weak Topology
- D. Bilinear Forms and Duality
- E. Hilbert Spaces: Fundamental Properties
- F. Hilbert Bases
- G. Historical Notes
- Chapter III Linearisation and Local Inversion of Differentiable Maps
- A. Differentiable Maps and Their Tangent Linear Maps
- B. The Chain Rule
- C. The Local Inversion Theorem
- D. Derivatives of Higher Order
- Part II THE GEOMETRIC SETTING
- Chapter IV Some Applications of Differential Calculus
- A. Geometric Variants of The Local Inversion Theorem
- B. Vector Fields and Ordinary Differential Equations
- C. Some Examples of Vector Fields
- D. Poisson Brackets and Conserved Quantities
- Chapter V A New Generalisation of the Notion of a Space: Configuration Spaces
- A. Local Coordinates and Configuration Spaces
- B. Differentiable Maps in Local Coordinates
- C. Vector Spaces in Curvilinear Coordinates
- D. The Fundamental Examples
- E. The Rotation Group
- F. Historical Notes
- Chapter VI Tangent Vectors and Vector Fields on Configuration Spaces
- A. Tangent Vectors to a Configuration Space
- B. Tangent Spaces to a Configuration Space
- C. Tangent Linear Maps to Differentiable Maps
- D. Vector Fields on Configuration Spaces
- E. Differential Equations on Configuration Spaces.
- F. Historical Notes
- Chapter VII Regular Points and Critical Points of Numerical Functions
- A. Differentials of Functions
- B. Submanifolds and Constraints
- C. Critical Points and Critical Values of Functions
- D. Hessians and Normal Forms at Generic Critical Points
- Part III THE CALCULUS OF VARIATIONS
- Chapter VIII Configuration Spaces of Geometric Objects
- A. Spaces of Curves
- B. Spaces of Surfaces in the Numerical Space
- C. On the Group of Diffeomorphisms
- D. Volume Elements
- E. Historical Notes (Contemporary)
- Chapter IX The Euler-Lagrange Equations
- A. The Extension by Velocities of a Configuration Space
- B. The Action and its First Variation
- C. The Euler-Lagrange Equations
- D. The Geometry of Geodesics
- E. Motion of a Rigid Body
- F. Surfaces of Stationary Area
- Chapter X The Hamiltonian Viewpoint
- A. The Extension by Momenta and its Symplectic Structure
- B. The General Form of Hamilton's Equations
- C. Relation with the Lagrangian Approach
- D. Poisson Brackets of Observables
- Chapter XI Symmetries and Conservation Laws
- A. Group Actions and Symmetries
- B. First Integrals and Conservation Laws
- C. The Notion of a Moment and the Theorem of E. Noether
- D. Observables in Involution and Integrable Systems
- Appendix: Basic Elements of Topology
- References
- Basic material on the topics of the notes
- Books supporting some complements to the notes
- More advanced books
- Books having a historical interest
- More recent books
- Notation Index
- Subject Index.
- Notes:
- Includes bibliographical references and index.
- Description based on print version record.
- Description based on publisher supplied metadata and other sources.
- Other Format:
- Print version: Bourguignon, Jean-Pierre Variational Calculus
- ISBN:
- 9783031183072
- OCLC:
- 1350791149
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