Variational Calculus with Engineering Applications
Description
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.
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Persons
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.
Content
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
1
Extrema of Differentiable Functionals
Motto:
"I wrote them with my nail on the plaster
On a wall of empty cracks,
In the dark, in my solitude,
Unaided by the bull lion vulture
of Luke, Mark and John."
Tudor Arghezi - Flowers of Mildew
Variational calculus aims to generalize the constructions of classical analysis, to solve extrema problems for functionals. In this introductory chapter, we will study the problem of differentiable functionals defined on various classes of functions. The news refers to path integral functionals. For details, see[ 4, 6, 7, 8, 14, 18, 21, 30, 58].
1.1 Differentiable Functionals
Let us consider the point and the function , . If this function has continuous partial derivatives with respect to each of the variables , then increasing the argument with produces
The first term on the right-hand side represents the differential of the function at the point , a linear form of argument growth, the vector ; the second term is the deviation from the linear approximation and is small in relation to , in the sense that
Let be two normed vector spaces. The previous definition can be extended immediately to the case of functions . If is a space of functions, and , then instead of function we use the term functional.
Definition 1.1 The function is called differentiable at a point if there exist a linear and continuous operator and a continuous function such that for any vector to have
The linear continuous operator is called the derivative of the function at given point .
For a given nonzero vector and , the vector
if the limit exists, is called the variation or derivative of the function f at the point x along the direction h.
A differentiable function of real variables has a derivative along any direction. The property is also kept for functions between normed vector spaces. Specifically, the next one takes place:
Proposition 1.1 If the function is differentiable at the point , then for any nonzero vector the function , of real variable , is derivable with respect to , for and
Proof. The derivative sought is obviously and then
Example 1.1: Any linear continuous operator is, obviously, a differentiable function at any point and since
Let be a functional. The simplest examples of functionals are given by formulas: (i) evaluation functional ("application of a function on the value at a point"), , where is a fixed point; (ii) definite integration (functional defined by definite integral); (iii) numerical quadrature defined by definite integration:
where means the set of all polynomials of degree at most ; and (iv) distributions in analysis (as linear functionals defined on spaces of test functions). We will continue to deal with definite integral type functionals, as the reasoning is more favorable to us.
Example 1.2: Let us consider the functional
defined on the space of the continuous functions on the segment , endowed with the norm of uniform convergence. The Lagrangian of the functional (i.e., the function ) is presumed continuous and with continuous partial derivatives of the first order in the domain .
Let us determine the variation of the functional when the argument increases with :
Since the Lagrangian is a differentiable function, we have
where
Therefore
In this way, according to the previous proposition and the derivation formula for integrals with parameters, we find
(1)Example 1.3: Let us consider the functional
defined on the space of functions with a continuous derivative on the segment , endowed with the norm of uniform convergence of derivatives. The Lagrangian of the functional is supposed to have first-order continuous partial derivatives. To write the integral functional, we use in fact the pullback form of the Lagrangian.
Applying the derivation formula of the integrals with parameter we obtain
(2)Analogously, we can extend the result to the functional
defined on the space of functions with continuous derivatives up to the order inclusive, on the segment . In this way we have
(3)Example 1.4: Let us consider a functional which depends on several function variables, for example
defined on the space , whose elements are vector functions , with the norm of uniform convergence of derivatives
( means the derivative with respect to ).
Supposing that the Lagrangian has continuous partial derivatives, denoting , the variation of the functional is
(4)(sum over the index ).
Now, let us consider functionals whose arguments are functions of several real variables.
Example 1.5: Let be a compact domain. As an example we will take, first, the double integral functional
where we noted for abbreviation
The functional is defined on the space of all functions with continuous partial derivatives; the norm of the space is given by
Supposing that the Lagrangian has continuous partial derivatives, the variation of the functional , as the argument grows with , is
(5)Example 1.6: Another example is the curvilinear integral functional
where is a piecewise curve which joins two fixed points in a compact domain .
Suppose the argument grows with . Then the variation of the functional is
(6)1.2 Extrema of Differentiable Functionals
Let us consider the functional , defined on a subset of a normed vector space of functions. By definition, the functional has a (relative) minimum at the point in , if there exists a neighborhood , of point , such that
If the point has a neighborhood on which the opposite inequality takes place
we say that is a point of local maximum for the functional . The minimum and maximum points are called relative extrema points.
In classical analysis, the extrema points of a differentiable function are among the critical points, that is, among the points that cancels first-order derivatives. A similar property occurs in the case of functional ones on normed vector spaces of functions, only in this case the extrema points are found between the extremals (critical points).
Proposition 1.2 If the function is an extremum point for the functional , interior point of the set and if is differentiable at this point, then for any growth .
Proof. Let be a growth of the argument (function); since is an interior point of the set , the function of real variable is defined on hole interval . This function has an extremum point at and is derivable at this point. Then its derivative must vanish at . It follows
Any point at which the variation of the functional is canceled identically with respect to is called either the stationary point or critical point or extremal of the functional.
Hence, for establishing the extrema points of a functional, the variation must be expressed, determine the critical points (those at which the...
