Variational calculus with engineering applications / Constantin Udriste and Ionel Tevy.

Includes bibliographical references and index.

Table of Contents

Preface ix



1 Extrema of Differentiable Functionals 1

1.1 Differentiable Functionals 1

1.2 Extrema of Differentiable Functionals 6

1.3 Second Variation; Sufficient Conditions for Extremum 14

1.4 Optimum with Constraints; the Principle of Reciprocity 17

1.4.1 Isoperimetric Problems 18

1.4.2 The Reciprocity Principle 19

1.4.3 Constrained Extrema: The Lagrange Problem 19

1.5 Maple Application Topics 21



2 Variational Principles 23

2.1 Problems with Natural Conditions at the Boundary 23

2.2 Sufficiency by the Legendre-Jacobi Test 27

2.3 Unitemporal Lagrangian Dynamics 30

2.3.1 Null Lagrangians 31

2.3.2 Invexity Test 32

2.4 Lavrentiev phenomenon 33

2.5 Unitemporal Hamiltonian Dynamics 35

2.6 Particular Euler–Lagrange ODEs 37

2.7 Multitemporal Lagrangian Dynamics 38

2.7.1 The Case of Multiple Integral Functionals 38

2.7.2 Invexity Test 40

2.7.3 The Case of Path-Independent Curvilinear Integral Functionals 41

2.7.4 Invexity Test 44

2.8 Multitemporal Hamiltonian Dynamics 45

2.9 Particular Euler–Lagrange PDEs 47

2.10 Maple Application Topics 48



3 Optimal Models Based on Energies 53

3.1 Brachistochrone Problem 53

3.2 Ropes, Chains and Cables 54

3.3 Newton’s Aerodynamic Problem 56

3.4 Pendulums 59

3.4.1 Plane Pendulum 59

3.4.2 Spherical Pendulum 60

3.4.3 Variable Length Pendulum 61

3.5 Soap Bubbles 62

3.6 Elastic Beam 63

3.7 The ODE of an Evolutionary Microstructure 63

3.8 The Evolution of a Multi-Particle System 64

3.8.1 Conservation of Linear Momentum 65

3.8.2 Conservation of Angular Momentum 66

3.8.3 Energy Conservation 67

3.9 String Vibration 67

3.10 Membrane Vibration 70

3.11 The Schrödinger Equation in Quantum Mechanics 73

3.11.1 Quantum Harmonic Oscillator 73

3.12 Maple Application Topics 74



4 Variational Integrators 79

4.1 Discrete Single-time Lagrangian Dynamics 79

4.2 Discrete Hamilton’s Equations 84

4.3 Numeric Newton’s Aerodynamic Problem 87

4.4 Discrete Multi-time Lagrangian Dynamics 88

4.5 Numerical Study of the Vibrating String Motion 92

4.5.1 Initial Conditions for Infinite String 94

4.5.2 Finite String, Fixed at the Ends 95

4.5.3 Monomial (Soliton) Solutions 96

4.5.4 More About Recurrence Relations 100

4.5.5 Solution by Maple via Eigenvalues 101

4.5.6 Solution by Maple via Matrix Techniques 102

4.6 Numerical Study of the Vibrating Membrane Motion 104

4.6.1 Monomial (Soliton) Solutions 105

4.6.2 Initial and Boundary Conditions 108

4.7 Linearization of Nonlinear ODEs and PDEs 109

4.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE 113

4.8.1 Von Neumann Analysis of Dual Variational Integrator Equation 115

4.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation 116

4.9 Maple Application Topics 119



5 Miscellaneous Topics 123

5.1 Magnetic Levitation 123

5.1.1 Electric Subsystem 123

5.1.2 Electromechanic Subsystem 124

5.1.3 State Nonlinear Model 124

5.1.4 The Linearized Model of States 125

5.2 The Problem of Sensors 125

5.2.1 Simplified Problem 126

5.2.2 Extending the Simplified Problem of Sensors 128

5.3 The Movement of a Particle in Non-stationary Gravito-vortex Field 128

5.4 Geometric Dynamics 129

5.4.1 Single-time Case 129

5.4.2 The Least Squares Lagrangian in Conditioning Problems 130

5.4.3 Multi-time Case 133

5.5 The Movement of Charged Particle in Electromagnetic Field 134

5.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential A 135

5.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction B 136

5.5.3 Unitemporal Geometric Dynamics Produced by Electric Field E 136

5.5.4 Potentials Associated to Electromagnetic Forms 137

5.5.5 Potential Associated to Electric 1-form E 138

5.5.6 Potential Associated to Magnetic 1-form H 138

5.5.7 Potential Associated to Potential 1-form A 138

5.6 Wind Theory and Geometric Dynamics 139

5.6.1 Pendular Geometric Dynamics and Pendular Wind 141

5.6.2 Lorenz Geometric Dynamics and Lorenz Wind 142

5.7 Maple Application Topics 143



6 Nonholonomic Constraints 147

6.1 Models With Holonomic and Nonholonomic Constraints 147

6.2 Rolling Cylinder as a Model with Holonomic Constraints 151

6.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint 152

6.3.1 Nonholonomic Geodesics 152

6.3.2 Geodesics in Sleigh Problem 155

6.3.3 Unicycle Dynamics 156

6.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot 157

6.5 Nonholonomic Constraints to the N-trailer 158

6.6 Famous Lagrangians 160

6.7 Significant Problems 160

6.8 Maple Application Topics 163



7 Problems: Free and Constrained Extremals 165

7.1 Simple Integral Functionals 165

7.2 Curvilinear Integral Functionals 169

7.3 Multiple Integral Functionals 171

7.4 Lagrange Multiplier Details 174

7.5 Simple Integral Functionals with ODE Constraints 175

7.6 Simple Integral Functionals with Nonholonomic Constraints 181

7.7 Simple Integral Functionals with Isoperimetric Constraints 184

7.8 Multiple Integral Functionals with PDE Constraints 186

7.9 Multiple Integral Functionals With Nonholonomic Constraints 188

7.10 Multiple Integral Functionals With Isoperimetric Constraints 189

7.11 Curvilinear Integral Functionals With PDE Constraints 191

7.12 Curvilinear Integral Functionals With Nonholonomic Constraints 193

7.13 Curvilinear Integral Functionals with Isoperimetric Constraints 195

7.14 Maple Application Topics 197

Bibliography 203

Index 209

"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.

About the Author

Constantin Udriste, PhD, is Professor Emeritus of Mathematics-Informatics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematics from the University Babes-Bolyai of Cluj-Napoca, Romania.

Ionel Tevy, PhD, is a Professor of Mathematics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematical Sciences from the Faculty of Mathematics and Mechanics of the University of Bucharest, Romania.