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Variational Calculus with Engineering Applications

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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 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.

Author(s): Constantin Udriste, Ionel Tevy
Publisher: Wiley
Year: 2022

Language: English
Pages: 223
City: Hoboken

Variational Calculus with Engineering Applications
Contents
Preface
1 Extrema of Differentiable Functionals
1.1 Differentiable Functionals
1.2 Extrema of Differentiable Functionals
1.3 Second Variation; Sufficient Conditions for Extremum
1.4 Optimum with Constraints; the Principle of Reciprocity
1.4.1 Isoperimetric Problems
1.4.2 The Reciprocity Principle
1.4.3 Constrained Extrema: The Lagrange Problem
1.5 Maple Application Topics
2 Variational Principles
2.1 Problems with Natural Conditions at the Boundary
2.2 Sufficiency by the Legendre-Jacobi Test
2.3 Unitemporal Lagrangian Dynamics
2.3.1 Null Lagrangians
2.3.2 Invexity Test
2.4 Lavrentiev Phenomenon
2.5 Unitemporal Hamiltonian Dynamics
2.6 Particular Euler–Lagrange ODEs
2.7 Multitemporal Lagrangian Dynamics
2.7.1 The Case of Multiple Integral Functionals
2.7.2 Invexity Test
2.7.3 The Case of Path-Independent Curvilinear Integral Functionals
2.7.4 Invexity Test
2.8 Multitemporal Hamiltonian Dynamics
2.9 Particular Euler–Lagrange PDEs
2.10 Maple Application Topics
3 Optimal Models Based on Energies
3.1 Brachistochrone Problem
3.2 Ropes, Chains and Cables
3.3 Newton’s Aerodynamic Problem
3.4 Pendulums
3.4.1 Plane Pendulum
3.4.2 Spherical Pendulum
3.4.3 Variable Length Pendulum
3.5 Soap Bubbles
3.6 Elastic Beam
3.7 The ODE of an Evolutionary Microstructure
3.8 The Evolution of a Multi-Particle System
3.8.1 Conservation of Linear Momentum
3.8.2 Conservation of Angular Momentum
3.8.3 Energy Conservation
3.9 String Vibration
3.10 Membrane Vibration
3.11 The Schrödinger Equation in Quantum Mechanics
3.11.1 Quantum Harmonic Oscillator
3.12 Maple Application Topics
4 Variational Integrators
4.1 Discrete Single-time Lagrangian Dynamics
4.2 Discrete Hamilton’s Equations
4.3 Numeric Newton’s Aerodynamic Problem
4.4 Discrete Multi-time Lagrangian Dynamics
4.5 Numerical Study of the Vibrating String Motion
4.5.1 Initial Conditions for Infinite String
4.5.2 Finite String, Fixed at the Ends
4.5.3 Monomial (Soliton) Solutions
4.5.4 More About Recurrence Relations
4.5.5 Solution by Maple via Eigenvalues
4.5.6 Solution by Maple via Matrix Techniques
4.6 Numerical Study of the Vibrating Membrane Motion
4.6.1 Monomial (Soliton) Solutions
4.6.2 Initial and Boundary Conditions
4.7 Linearization of Nonlinear ODEs and PDEs
4.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE
4.8.1 Von Neumann Analysis of Dual Variational Integrator Equation
4.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation
4.9 Maple Application Topics
5 Miscellaneous Topics
5.1 Magnetic Levitation
5.1.1 Electric Subsystem
5.1.2 Electromechanic Subsystem
5.1.3 State Nonlinear Model
5.1.4 The Linearized Model of States
5.2 The Problem of Sensors
5.2.1 Simplified Problem
5.2.2 Extending the Simplified Problem of Sensors
5.3 The Movement of a Particle in Non-stationary Gravito-vortex Field
5.4 Geometric Dynamics
5.4.1 Single-time Case
5.4.2 The Least Squares Lagrangian in Conditioning Problems
5.4.3 Multi-time Case
5.5 The Movement of Charged Particle in Electromagnetic Field
5.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential
5.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction
5.5.3 Unitemporal Geometric Dynamics Produced by Electric Field
5.5.4 Potentials Associated to Electromagnetic Forms
5.5.5 Potential Associated to Electric 1-form
5.5.6 Potential Associated to Magnetic 1-form
5.5.7 Potential Associated to Potential 1-form
5.6 Wind Theory and Geometric Dynamics
5.6.1 Pendular Geometric Dynamics and Pendular Wind
5.6.2 Lorenz Geometric Dynamics and Lorenz Wind
5.7 Maple Application Topics
6 Nonholonomic Constraints
6.1 Models With Holonomic and Nonholonomic Constraints
6.2 Rolling Cylinder as a Model with Holonomic Constraints
6.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint
6.3.1 Nonholonomic Geodesics
6.3.2 Geodesics in Sleigh Problem
6.3.3 Unicycle Dynamics
6.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot
trailer
6.5 Nonholonomic Constraints to the
6.6 Famous Lagrangians
6.7 Significant Problems
6.8 Maple Application Topics
7 Problems: Free and Constrained Extremals
7.1 Simple Integral Functionals
7.2 Curvilinear Integral Functionals
7.3 Multiple Integral Functionals
7.4 Lagrange Multiplier Details
7.5 Simple Integral Functionals with ODE Constraints
7.6 Simple Integral Functionals with Nonholonomic Constraints
7.7 Simple Integral Functionals with Isoperimetric Constraints
7.8 Multiple Integral Functionals with PDE Constraints
7.9 Multiple Integral Functionals With Nonholonomic Constraints
7.10 Multiple Integral Functionals With Isoperimetric Constraints
7.11 Curvilinear Integral Functionals With PDE Constraints
7.12 Curvilinear Integral Functionals With Nonholonomic Constraints
7.13 Curvilinear Integral Functionals with Isoperimetric Constraints
7.14 Maple Application Topics
Bibliography
Index
EULA