Variational calculus with elementary convexity

John L. Troutman ; with the assistance of W. Hrusa

[目次]

  • 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 Brachistochrone.- (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 Gateaux Variations.- 2.1. Real Linear Spaces.- 2.2. Functions from Linear Spaces.- 2.3. Fundamentals of Optimization.- Constraints.- Rotating Fluid Column.- 2.4. The Gateaux 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 Brachistochrone.- (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 Frechet 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 Brachistochrone.- 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 Brachistochrone*.- 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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この本の情報

書名 Variational calculus with elementary convexity
著作者等 Hrusa, W.
Troutman, John L.
Hrusa W.
Troutman J. L.
シリーズ名 Undergraduate texts in mathematics
出版元 Springer-Verlag
刊行年月 c1983
版表示 Softcover reprint of the original 1st ed. 1983
ページ数 xiv, 364 p.
大きさ 25 cm
ISBN 3540907718
0387907718
9781468401608
NCID BA01308468
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
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