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 Applications 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.
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
