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| Introduction | |
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| Introduction | |
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| The Catenary and Brachystochrone Problems | |
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| The Catenary | |
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| Brachystochrones | |
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| Hamilton's Principle | |
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| Some Variational Problems from Geometry | |
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| Dido's Problem | |
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| Geodesics | |
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| Minimal Surfaces | |
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| Optimal Harvest Strategy | |
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| The First Variation | |
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| The Finite-Dimensional Case | |
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| Functions of One Variable | |
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| Functions of Several Variables | |
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| The Euler-Lagrange Equation | |
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| Some Special Cases | |
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| Case I: No Explicity y Dependence | |
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| Case II: No Explicit x Dependence | |
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| A Degenerate Case | |
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| Invariance of the Euler-Lagrange Equation | |
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| Existence of Solutions to the Boundary-Value Problem* | |
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| Some Generalizations | |
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| Functionals Containing Higher-Order Derivatives | |
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| Several Dependent Variables | |
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| Two Independent Variables* | |
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| The Inverse Problem* | |
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| Isoperimetric Problems | |
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| The Finite-Dimensional Case and Lagrange Multipliers | |
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| Single Constraint | |
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| Multiple Constraints | |
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| Abnormal Problems | |
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| The Isoperimetric Problem | |
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| Some Generalizations on the Isoperimetric Problem | |
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| Problems Containing Higher-Order Derivatives | |
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| Multiple Isoperimetric Constraints | |
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| Several Dependent Variables | |
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| Applications to Eigenvalue Problems* | |
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| The Sturm-Liouville Problem | |
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| The First Eigenvalue | |
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| Higher Eigenvalues | |
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| Holonomic and Nonholonomic Constraints | |
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| Holonomic Constraints | |
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| Nonholonomic Constraints | |
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| Nonholonomic Constraints in Mechanics* | |
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| Problems with Variable Endpoints | |
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| Natural Boundary Conditions | |
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| The General Case | |
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| Transversality Conditions | |
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| The Hamiltonian Formulation | |
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| The Legendre Transformation | |
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| Hamilton's Equations | |
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| Symplectic Maps | |
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| The Hamilton-Jacobi Equation | |
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| The General Problem | |
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| Conservative Systems | |
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| Separation of Variables | |
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| The Method of Additive Separation | |
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| Conditions for Separable Solutions* | |
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| Noether's Theorem | |
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| Conservation Laws | |
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| Variational Symmetries | |
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| Noether's Theorem | |
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| Finding Variational Symmetries | |
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| The Second Variation | |
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| The Finite-Dimensional Case | |
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| The Second Variation | |
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| The Legendre Condition | |
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| The Jacobi Necessary Condition | |
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| A Reformulation of the Second Variation | |
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| The Jacobi Accessory Equation | |
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| The Jacobi Necessary Condition | |
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| A Sufficient Condition | |
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| More on Conjugate Points | |
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| Finding Conjugate Points | |
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| A Geometrical Interpretation | |
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| Saddle Points* | |
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| Convex Integrands | |
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| Analysis and Differential Equations | |
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| Taylor's Theorem | |
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| The Implicit Function Theorem | |
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| Theory of Ordinary Differential Equations | |
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| Function Spaces | |
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| Normed Spaces | |
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| Banach and Hilbert Spaces | |
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| References | |
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| Index | |