Variational Calculus With Elementary Convexity

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