| 1 Introduction 1.1 Introduction 1.2 The Catenary and Brachystochrone Problems 1.2.1 The Catenary 1.2.2 Brachystochrones 1.3 Hamilton's Principle 1.4 Some Variational Problems from Geometry 1.4.1 Dido's Problem 1.4.2 Geodesics 1.4.3 Minimal Surfaces 1.5 Optimal Harvest Strategy 2 The First Variation 2.1 The Finite-Dimensional Case 2.1.1 Functions of One Variable 2.1.2 Functions of Several Variables 2.2 The Euler-Lagrange Equation 2.3 Some Special Cases 2.3.1 Case I: No Explicit y Dependence 2.3.2 Case II: No Explicit x Dependence 2.4 A Degenerate Case 2.5 Invariance of the Euler-Lagrange Equation 2.6 Existence of Solutions to the Boundary-Value Problem 3 Some Generalizations 3.1 Functionals Containing Higher-Order Derivatives 3.2 Several Dependent Variables 3.3 Two Independent Variables 3.4 The Inverse Problem 4 Isoperimetric Problems 4.1 The Finite-Dimensional Case and Lagrange Multipliers 4.1.1 Single Constraint 4.1.2 Multiple Constraints 4.1.3 Abnormal Problems 4.2 The Isoperimetric Problem 4.3 Some Generalizations on the Isoperimetric Problem 4.3.1 Problems Containing Higher-Order Derivatives 4.3.2 Multiple Isoperimetric Constraints 4.3.3 Several Dependent Variables 5 Applications to Eigenvalue Problems 5.1 The Sturm-Liouville Problem 5.2 The First Eigenvalue 5.3 Higher Eigenvalues 6 Holonomic and Nonholonomic Constraints 6.1 Holonomic Constraints 6.2 Nonholonomic Constraints 6.3 Nonholonomic Constraints in Mechanics 7 Problems with Variable Endpoints 7.1 Natural Boundary Conditions 7.2 The General Case 7.3 Tansversality Conditions 8 The Hamiltonin Formulation 8.1 The Legendre Transformation 8.2 Hamilton's Equations 8.3 Symplectic Maps 8.4 The Hamilton-Jacobi Equation 8.4.1 The General Problem 8.4.2 Conservative Systems 8.5 Separation of Variables 8.5.1 The Method of Additive Separation 8.5.2 Conditions for Separable Solutions 9 Noether's Theorem 9.1 Conservation Laws 9.2 Variational Symmetries 9.3 Noether's Theorem 9.4 Finding Varbational Symmetries 10 The Second Variation 10.1 The Finite-Dimensional Case 10.2 The Second Variation 10.3 The Legendre Condition 10.4 The Jacobi Necessary Condition 10.4.1 A Reformulation of the Second Variation 10.4.2 The Jacobi Accessory Equation 10.4.3 The Jacobi Necessary Condition 10.5 A Sufficient Condition 10.6 More on Conjugate Points 10.6.1 Finding Conjugate Points 10.6.2 A Geometrical Interpretation 10.6.3 Saddle Points 10.7 Convex Integrands A Analysis and Differential Equations A.1 Taylor's Theorem A.2 The Implicit Function Theorem A.3 Theory of Ordinary Differential Equations B Function Spaces B.1 Normed Spaces B.2 Banach and Hilbert Spaces References Index |
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