The Inverse Problem of the Calculus of Variations: Local and Global Theory (Atlantis Studies in Variational Geometry Book 2)

The Aim Of The Present Book Is To Give A Systematic Treatment Of The Inverse Problem Of The Calculus Of Variations, I.e. How To Recognize Whether A System Of Differential Equations Can Be Treated As A System For Extremals Of A Variational Functional (the Euler-lagrange Equations), Using Contemporary Geometric Methods. Selected Applications In Geometry, Physics, Optimal Control, And General Relativity Are Also Considered. The Book Includes The Following Chapters: - Helmholtz Conditions And The Method Of Controlled Lagrangians (bloch, Krupka, Zenkov) - The Sonin-douglas's Problem (krupka) - Inverse Variational Problem And Symmetry In Action: The Ostrogradskyj Relativistic Third Order Dynamics (matsyuk.) - Source Forms And Their Variational Completion (voicu) - First-order Variational Sequences And The Inverse Problem Of The Calculus Of Variations (urban, Volna) - The Inverse Problem Of The Calculus Of Variations On Grassmann Fibrations (urban). The Helmholtz Conditions And The Method Of Controlled Lagrangians -- The Sonin–douglas Problem -- Inverse Variational Problem And Symmetry In Action: The Relativistic Third Order Dynamics -- Variational Principles For Immersed Submanifolds -- Source Forms And Their Variational Completions -- First-order Variational Sequences In Field Theory. Edited By Dmitry V. Zenkov. Preface 6 Contents 8 Contributors 9 1 The Helmholtz Conditions and the Method of Controlled Lagrangians 10 1.1 Introduction 10 1.2 The Helmholtz Conditions 13 1.3 Underactuated Systems 15 1.3.1 Controlled Vector Fields 16 1.3.2 Matching and Controlled Lagrangians 16 1.3.3 Examples 20 1.3.4 Generalized Matching 27 1.4 Controlled Lagrangians and the Inverse Problem of the Calculus of Variations 30 1.4.1 The Helmholtz Conditions as Generalized Matching Conditions 30 1.4.2 The Inverted Pendulum on a Cart Revisited 31 1.4.3 The Helmholtz Conditions and Linear Lagrangian Systems 32 1.4.4 Stabilization by Controlled Lagrangians 34 References 36 2 The Sonin--Douglas Problem 39 2.1 Introduction 39 2.2 Energy Lagrangians 41 2.3 Integrability Conditions 45 2.4 Variational Systems and the Helmholtz Conditions 47 2.5 The Structure of Variational Systems 53 2.6 The Sonin--Douglas Problem 64 2.7 Finsler Metrics 73 References 81 3 Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics 82 3.1 Introduction 82 3.2 Homogeneous Form and Parametric Invariance 84 3.3 The Criterion of Variationality 87 3.4 The Lepagean Equivalent 90 3.5 The Invariant Euler--Poisson Equation 91 3.6 An Example: Free Relativistic Top in Two Timensions 93 3.7 The Inverse Problem for the Euler--Poisson Equations 93 3.7.1 The Generalized Helmholtz Conditions 93 3.7.2 The Fourth Order Variational ODEs 97 3.7.3 The Third Order Variational ODEs 100 3.7.4 Remarks on the First and Second Order Variational ODEs 101 3.7.5 mathbbE4: The No-go Theorem 101 3.7.6 The Invariant Euler--Poisson Equation of a Relativistic Two-Dimensional Motion 102 References 108 4 Variational Principles for Immersed Submanifolds 110 4.1 Introduction 110 4.2 Higher-Order Grassmann Fibrations 113 4.2.1 Manifolds of Velocities 113 4.2.2 Regular Velocities 117 4.2.3 Grassmann Fibrations 120 4.3 Contact Differential Forms 124 4.3.1 Contact Forms on Grassmann Fibrations 124 4.3.2 Contact Decomposition of Forms 127 4.3.3 Lepage One-Forms on Grassmann Fibrations 132 4.4 Variational Principles on Grassmann Fibrations: Lepage Forms 134 4.4.1 Variational Functionals on Grassmann Fibrations 135 4.4.2 The Noether Theorem for Submanifolds 141 4.4.3 Example: Extremals on a Sphere 144 4.5 The Inverse Variational Problem: Variational Sequences 147 4.5.1 Variational Sequences on Grassmann Fibrations 147 4.5.2 Classes of Forms in the Variational Sequence 149 4.5.3 The Inverse Problem for Second-Order Systems 159 4.6 Homogeneous Systems on Manifolds of Velocities 163 4.6.1 Higher-Order Homogeneous Functions 164 4.6.2 Invariant Variational Functionals 168 4.6.3 Higher-Order Homogeneous and Variational Systems 170 References 176 5 Source Forms and Their Variational Completions 178 5.1 Introduction 178 5.2 Preliminaries 179 5.2.1 Differential Forms on Jet Prolongations of a Fibered Manifold 179 5.2.2 Lepage Equivalent of a Lagrangian 182 5.3 Source Forms and Variationality Conditions 187 5.3.1 Basic Results 187 5.3.2 First-Order Variational Forms 188 5.4 Variational Completions of a Source Form 194 5.4.1 Canonical Variational Completion 195 5.4.2 Lower-Order Variational Completions 198 5.5 Source Forms in General Relativity 203 5.5.1 The Hilbert Action and the Vacuum Einstein Equation: A Differential Form Approach 204 5.5.2 The Einstein Equations With Matter and Energy-Momentum Tensors 209 5.5.3 Energy-Momentum Conservation and Bianchi Identities 211 5.5.4 Canonical Variational Completions 214 5.5.5 Classification of First-Order Energy-Momentum Tensors 219 References 220 6 First-Order Variational Sequences in Field Theory 222 6.1 Introduction 222 6.2 Jet Prolongations of Fibred Manifolds 227 6.3 Contact Differential Forms 231 6.4 Variational Theory on Fibred Manifolds 237 6.4.1 Lepage Forms 237 6.4.2 The First Variation Formula 243 6.4.3 Extremals 245 6.5 Variational Sequences 246 6.5.1 Variational Sequences on Finite-Order Jet Spaces 246 6.5.2 Representation of Variational Sequences: n-forms, (n+1)-forms, and (n+2)-forms 249 6.5.3 Explicit Expressions of Classes of Differential Forms 254 6.6 The Global Inverse Variational Problem 262 6.6.1 The Euler--Lagrange Mapping 262 6.6.2 The Helmholtz Form 269 6.6.3 Local and Global Variationality 269 6.7 Invariant Forms and the Noether Theorem 270 6.7.1 Invariant Lagrangians 271 6.7.2 Invariant Euler--Lagrange Forms 275 6.7.3 Inverse Variational Problem for Invariant Source Forms 276 6.8 Variational Principles in General Relativity: Examples 279 6.8.1 The Hilbert Variational Principle 279 6.8.2 Interaction of Gravitational and Electromagnetic Fields 287 References 288 Index 292 Front Matter....Pages i-ix The Helmholtz Conditions and the Method of Controlled Lagrangians....Pages 1-29 The Sonin–Douglas Problem....Pages 31-73 Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics....Pages 75-102 Variational Principles for Immersed Submanifolds....Pages 103-170 Source Forms and Their Variational Completions....Pages 171-214 First-Order Variational Sequences in Field Theory....Pages 215-284 Back Matter....Pages 285-289