Variational Calculus with Engineering Applications

VARIATIONAL CALCULUS WITH ENGINEERING APPLICATIONS A comprehensive overview of foundational variational methods for problems in engineering Variational calculus is a field in which small alterations in functions and functionals are used to find their relevant maxima and minima. It is a potent tool for addressing a range of dynamic problems with otherwise counter-intuitive solutions, particularly ones incorporating multiple confounding variables. Its value in engineering fields, where materials and geometric configurations can produce highly specific problems with unconventional or unintuitive solutions, is considerable. Variational Calculus with Engineering Applications provides a comprehensive survey of this toolkit and its engineering applications. Balancing theory and practice, it offers a thorough and accessible introduction to the field pioneered by Euler, Lagrange and Hamilton, offering tools that can be every bit as powerful as the better-known Newtonian mechanics. It is an indispensable resource for those looking for engineering-oriented overview of a subject whose capacity to provide engineering solutions is only increasing. Variational Calculus with Engineering Application s readers will also find: Discussion of subjects including variational principles, levitation, geometric dynamics, and more Examples and instructional problems in every chapter, along with MAPLE codes for performing the simulations described in each Engineering applications based on simple, curvilinear, and multiple integral functionals Variational Calculus with Engineering Applications is ideal for advanced students, researchers, and instructors in engineering and materials science. Variational Calculus with Engineering Applications 2 Contents 6 Preface 10 1 Extrema of Differentiable Functionals 12 1.1 Differentiable Functionals 12 1.2 Extrema of Differentiable Functionals 17 1.3 Second Variation; Sufficient Conditions for Extremum 25 1.4 Optimum with Constraints; the Principle of Reciprocity 28 1.4.1 Isoperimetric Problems 29 1.4.2 The Reciprocity Principle 30 1.4.3 Constrained Extrema: The Lagrange Problem 30 1.5 Maple Application Topics 32 2 Variational Principles 34 2.1 Problems with Natural Conditions at the Boundary 34 2.2 Sufficiency by the Legendre-Jacobi Test 38 2.3 Unitemporal Lagrangian Dynamics 41 2.3.1 Null Lagrangians 42 2.3.2 Invexity Test 43 2.4 Lavrentiev Phenomenon 44 2.5 Unitemporal Hamiltonian Dynamics 46 2.6 Particular Euler–Lagrange ODEs 48 2.7 Multitemporal Lagrangian Dynamics 49 2.7.1 The Case of Multiple Integral Functionals 49 2.7.2 Invexity Test 51 2.7.3 The Case of Path-Independent Curvilinear Integral Functionals 52 2.7.4 Invexity Test 55 2.8 Multitemporal Hamiltonian Dynamics 56 2.9 Particular Euler–Lagrange PDEs 58 2.10 Maple Application Topics 59 3 Optimal Models Based on Energies 64 3.1 Brachistochrone Problem 64 3.2 Ropes, Chains and Cables 66 3.3 Newton’s Aerodynamic Problem 67 3.4 Pendulums 70 3.4.1 Plane Pendulum 70 3.4.2 Spherical Pendulum 71 3.4.3 Variable Length Pendulum 72 3.5 Soap Bubbles 73 3.6 Elastic Beam 74 3.7 The ODE of an Evolutionary Microstructure 74 3.8 The Evolution of a Multi-Particle System 75 3.8.1 Conservation of Linear Momentum 76 3.8.2 Conservation of Angular Momentum 77 3.8.3 Energy Conservation 78 3.9 String Vibration 78 3.10 Membrane Vibration 81 3.11 The Schrödinger Equation in Quantum Mechanics 84 3.11.1 Quantum Harmonic Oscillator 84 3.12 Maple Application Topics 85 4 Variational Integrators 90 4.1 Discrete Single-time Lagrangian Dynamics 90 4.2 Discrete Hamilton’s Equations 95 4.3 Numeric Newton’s Aerodynamic Problem 98 4.4 Discrete Multi-time Lagrangian Dynamics 99 4.5 Numerical Study of the Vibrating String Motion 103 4.5.1 Initial Conditions for Infinite String 105 4.5.2 Finite String, Fixed at the Ends 106 4.5.3 Monomial (Soliton) Solutions 107 4.5.4 More About Recurrence Relations 111 4.5.5 Solution by Maple via Eigenvalues 112 4.5.6 Solution by Maple via Matrix Techniques 113 4.6 Numerical Study of the Vibrating Membrane Motion 115 4.6.1 Monomial (Soliton) Solutions 116 4.6.2 Initial and Boundary Conditions 119 4.7 Linearization of Nonlinear ODEs and PDEs 120 4.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE 124 4.8.1 Von Neumann Analysis of Dual Variational Integrator Equation 126 4.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation 127 4.9 Maple Application Topics 130 5 Miscellaneous Topics 134 5.1 Magnetic Levitation 134 5.1.1 Electric Subsystem 134 5.1.2 Electromechanic Subsystem 135 5.1.3 State Nonlinear Model 135 5.1.4 The Linearized Model of States 136 5.2 The Problem of Sensors 136 5.2.1 Simplified Problem 137 5.2.2 Extending the Simplified Problem of Sensors 139 5.3 The Movement of a Particle in Non-stationary Gravito-vortex Field 139 5.4 Geometric Dynamics 140 5.4.1 Single-time Case 140 5.4.2 The Least Squares Lagrangian in Conditioning Problems 141 5.4.3 Multi-time Case 144 5.5 The Movement of Charged Particle in Electromagnetic Field 145 5.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential 146 5.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction 147 5.5.3 Unitemporal Geometric Dynamics Produced by Electric Field 147 5.5.4 Potentials Associated to Electromagnetic Forms 148 5.5.5 Potential Associated to Electric 1-form 149 5.5.6 Potential Associated to Magnetic 1-form 149 5.5.7 Potential Associated to Potential 1-form 149 5.6 Wind Theory and Geometric Dynamics 150 5.6.1 Pendular Geometric Dynamics and Pendular Wind 152 5.6.2 Lorenz Geometric Dynamics and Lorenz Wind 153 5.7 Maple Application Topics 154 6 Nonholonomic Constraints 158 6.1 Models With Holonomic and Nonholonomic Constraints 158 6.2 Rolling Cylinder as a Model with Holonomic Constraints 162 6.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint 163 6.3.1 Nonholonomic Geodesics 163 6.3.2 Geodesics in Sleigh Problem 166 6.3.3 Unicycle Dynamics 167 6.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot 168 trailer 169 6.5 Nonholonomic Constraints to the 169 6.6 Famous Lagrangians 171 6.7 Significant Problems 171 6.8 Maple Application Topics 174 7 Problems: Free and Constrained Extremals 176 7.1 Simple Integral Functionals 176 7.2 Curvilinear Integral Functionals 180 7.3 Multiple Integral Functionals 182 7.4 Lagrange Multiplier Details 185 7.5 Simple Integral Functionals with ODE Constraints 186 7.6 Simple Integral Functionals with Nonholonomic Constraints 192 7.7 Simple Integral Functionals with Isoperimetric Constraints 195 7.8 Multiple Integral Functionals with PDE Constraints 197 7.9 Multiple Integral Functionals With Nonholonomic Constraints 199 7.10 Multiple Integral Functionals With Isoperimetric Constraints 200 7.11 Curvilinear Integral Functionals With PDE Constraints 202 7.12 Curvilinear Integral Functionals With Nonholonomic Constraints 204 7.13 Curvilinear Integral Functionals with Isoperimetric Constraints 206 7.14 Maple Application Topics 208 Bibliography 214 Index 220 EULA 224 "The Variational Calculus with Engineering Applications was and is being taught to 4th year engineering students, in the Faculty of Applied Sciences, Mathematics - Informatics Department, from the University Politehnica of Bucharest, by Prof. Emeritus Dr. Constantin Udriste. Certain topics are taught at other faculties of our university, especially at master's or doctoral courses, being present in the papers that can be published in journals now categorized as "ISI". The Chapters were structured according to the importance, accessibility and impact of the theoretical notions able to outline a future specialist based on mathematical optimization tools. The probing and intermediate variants lasted for a number of sixteen years, leading to the selection of the most important manageable notions and reaching maturity through this variant that we decided to publish at Wiley. Now the topics of the book includes seven Chapters: Extrema of Dierentiable Functionals; Variational Principles; Optimal Models Based on Energies; Variational Integrators; Miscellaneous Topics; Extremals with Nonholonomic Constraints; Problems: Free and Constrained Extremals. To cover modern problem-solving methods, each Chapter includes Maple application topics."-- Provided by publisher