Variational Calculus with Elementary Convexity

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Product Details

  • ISBN-13: 9781468401608
  • Publisher: Springer New York
  • Publication date: 2/2/2012
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: Softcover reprint of the original 1st ed. 1983
  • Edition number: 1
  • Pages: 384
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.79 (d)

Table of Contents

0 Review of Optimization in—d.- Problems.- One Basic Theory.- 1 Standard Optimization Problems.- 1.1. Geodesic Problems.- (a) Geodesics in—d.- (b) Geodesics on a Sphere.- (c) Other Geodesic Problems.- 1.2. Time-of-Transit Problems.- (a) The Brachishrone.- (b) Steering Problems.- 1.3. Isoperimetric Problems.- 1.4. Surface Area Problems.- (a) Minimal Surface of Revolution.- (b) Minimal Area Problem.- (c) Plateau’s Problem.- 1.5. Summary: Plan of the Text.- Notation: Uses and Abuses.- Problems.- 2 Linear Spaces and Gâteaux Variations.- 2.1. Real Linear Spaces.- 2.2. Functions from Linear Spaces.- 2.3. Fundamentals of Optimization.- Constraints.- Rotating Fluid Column.- 2.4. The Gâteaux Variations.- Problems.- 3 Minimization of Convex Functions.- 3.1. Convex Functions.- 3.2. Convex Integral Functions.- Free End Point Problems.- 3.3. [Strongly] Convex Functions.- 3.4. Applications.- (a) Geodesics on a Cylinder.- (b) A Brachishrone.- (c) A Profile of Minimum Drag.- (d) An Economics Problem.- (e) Minimal Area Problem.- 3.5. Minimization with Convex Constraints.- The Hanging Cable.- Optimal Performance.- 3.6. Summary: Programs for Minimization.- Problems.- 4 The Lemmas of Lagrange and Du Bois-Reymond.- Problems.- 5 Local Extrema in Normed Linear Spaces.- 5.1. Norms for Linear Spaces.- 5.2. Normed Linear Spaces: Convergence and Compactness.- 5.3. Continuity.- 5.4. (Local) Extremal Points.- 5.5. Necessary Conditions: Admissible Directions.- 5.6*. Affine Approximation: The Fréchet Derivative.- Tangency.- 5.7. Extrema with Constraints: Lagrangian Multipliers.- Problems.- 6 The Euler-Lagrange Equations.- 6.1. The First Equation: Stationary Functions.- 6.2. Special Cases of the First Equation.- (a) When f = f(z).- (b) When f = f(x, z).- (c) When f = f(y, z).- 6.3. The Second Equation.- 6.4. Variable End Point Problems: Natural Boundary Conditions.- Jakob Bernoulli’s Brachishrone.- Transversal Conditions*.- 6.5. Integral Constraints: Lagrangian Multipliers.- 6.6. Integrals Involving Higher Derivatives.- Buckling of a Column under Compressive Load.- 6.7. Vector Valued Stationary Functions.- The Isoperimetric Problem.- Lagrangian Constraints*.- Geodesics on a Surface.- 6.8*. Invariance of Stationarity.- 6.9. Multidimensional Integrals.- Minimal Area Problem.- Natural Boundary Conditions.- Problems.- Two Advanced Topics.- 7 Piecewise C1 Extremal Functions.- 7.1. Piecewise C1 Functions.- (a) Smoothing.- (b) Norms for—1.- 7.2. Integral Functions on—1.- 7.3. Extremals in—1 [a, b]: The Weierstrass-Erdmann Corner Conditions.- A Sturm-Liouville Problem.- 7.4. Minimization Through Convexity.- Internal Constraints.- 7.5. Piecewise C1 Vector Valued Extremals.- Minimal Surface of Revolution.- Hilbert’s Differentiability Criterion*.- 7.6*. Conditions Necessary for a Local Minimum.- (a) The Weierstrass Condition.- (b) The Legendre Condition.- Bolza’s Problem.- Problems.- 8 Variational Principles in Mechanics.- 8.1. The Action Integral.- 8.2. Hamilton’s Principle: Generalized Coordinates.- Bernoulli’s Principle of Static Equilibrium.- 8.3. The Total Energy.- Spring-Mass-Pendulum System.- 8.4. The Canonical Equations.- 8.5. Integrals of Motion in Special Cases.- Jacobi’s Principle of Least Action.- Symmetry and Invariance.- 8.6. Parametric Equations of Motion.- 8.7*. The Hamilton-Jacobi Equation.- 8.8. Stationary Functions; Complementary Inequalities.- 8.9. Continuous Media.- (a) Taut String.- The Nonuniform String.- (b) Stretched Membrane.- Static Equilibrium of (Nonplanar) Membrane.- Problems.- 9* Sufficient Conditions for a Minimum.- 9.1. The Weierstrass Method.- 9.2. [Strict] Convexity of f(x, Y, Z).- 9.3. Fields.- Exact Fields and the Hamilton-Jacobi Equation*.- 9.4. Hilbert’s Invariant Integral.- The Brachishrone*.- Variable End Point Problems.- 9.5. Minimization with Constraints.- The Wirtinger Inequality.- 9.6.* Central Fields.- Smooth Minimal Surface of Revolution.- 9.7. Construction of Central Fields with Given Trajectory; The Jacobi Condition.- 9.8. Sufficient Conditions for a Local Minimum.- (a) Pointwise Results.- Hamilton’s Principle.- (b) Trajectory Results.- 9.9*. Necessity of the Jacobi Condition.- 9.10. Concluding Remarks.- Problems.- A.1. The Intermediate and Mean Value Theorems.- A.2. The Fundamental Theorem of Calculus.- A.3. Partial Integrals: Leibniz’ Formula.- A.4. An Open Mapping Theorem.- A.5. Families of Solutions to a System of Differential Equations.- A.6. The Rayleigh Ratio.- Answers to Selected Problems.

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