Variational Calculus

Description

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. Stressing the need to overcome limitations of the initial point of view, and emphasising the interconnectivity of various branches of mathematics (algebra, analysis and geometry), the book includes some advanced material to challenge the most motivated students. Systematic, short historical notes provide details on the subject’s odyssey, and how new tools have been developed over the last two centuries. This English translation updates a set of notes for a course first given at the École polytechnique in 1987. It will be accessible to graduate students and advanced undergraduates. 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 E. Historical Notes 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 E. Historical Notes 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 E. Historical Notes 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 G. Historical Notes 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 E. Historical Notes 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 E. Historical Notes 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