book

Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

Name: Applied Calculus of Variations for Engineers, Third edition, 3rd Edition
Author: Louis Komzsik
ISBN: 9781000764758

by Louis Komzsik

November 2019

Intermediate to advanced

291 pages

4h 21m

English

CRC Press

Read now

Unlock full access

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Acknowledgments
Author
Introduction

Part I: Mathematical foundation
1. The foundations of calculus of variations
1.1 The fundamental problem and lemma of calculus of variations1.2 The Legendre test1.3 The Euler–Lagrange differential equation1.4 Minimal path problems1.4.1 Shortest curve between two points1.4.2 The brachistochrone problem1.4.3 Fermat’s principle1.4.4 Particle moving in a gravitational field1.5 Open boundary variational problems1.6 Exercises
2. Constrained variational problems
2.1 Algebraic boundary conditions2.1.1 Transversality condition computation2.2 Lagrange’s solution2.3 Isoperimetric problems2.3.1 Maximal area under curve with given length2.3.2 Optimal shape of curve of given length under gravity2.4 Closed-loop integrals2.5 Exercises
3. Multivariate functionals
3.1 Functionals with several functions3.1.1 Euler–Lagrange system of equations3.2 Variational problems in parametric form3.2.1 Maximal area by closed parametric curve3.3 Functionals with two independent variables3.4 Minimal surfaces3.4.1 Minimal surfaces of revolution3.5 Functionals with three independent variables3.6 Exercises
4. Higher order derivatives
4.1 The Euler–Poisson equation4.2 The Euler–Poisson system of equations4.3 Algebraic constraints on the derivative4.4 Linearization of second order problems4.5 Exercises
5. The inverse problem
5.1 Linear differential operators5.2 The variational form of Poisson’s equation5.3 The variational form of eigenvalue problems5.3.1 Orthogonal eigensolutions5.4 Sturm–Liouville problems5.4.1 Legendre’s equation and polynomials5.5 Exercises
6. Analytic solutions
6.1 Laplace transform solution6.2 d’Alembert’s solution6.3 Complete integral solutions6.4 Poisson’s integral formula6.5 Method of gradients6.6 Exercises
7. Approximate methods
7.1 Euler’s method7.2 Ritz’s method7.3 Galerkin’s method7.4 Approximate solutions of Poisson’s equation7.5 Kantorovich’s method7.6 Boundary integral method7.7 Finite element method7.8 Exercises
Part II: Modeling applications
8. Differential geometry
8.1 The geodesic problem8.1.1 Geodesics of a sphere8.1.2 Geodesic polyhedra8.2 A system of differential equations for geodesic curves8.2.1 Geodesics of surfaces of revolution8.3 Geodesic curvature8.3.1 Geodesic curvature of helix8.4 Generalization of the geodesic concept
9. Computational geometry
9.1 Natural splines9.2 B-spline approximation9.3 B-splines with point constraints9.4 B-splines with tangent constraints9.5 Generalization to higher dimensions9.6 Weighting and non-uniform parametrization9.7 Industrial applications
10. Variational equations of motion
10.1 Legendre’s dual transformation10.2 Hamilton’s principle10.3 Hamilton’s canonical equations10.3.1 Conservation of energy10.3.2 Newton’s law of motion10.4 Lagrange’s equations of motion10.4.1 Mechanical system modeling10.4.2 Electro-mechanical analogy10.5 Orbital motion10.5.1 Conservation of angular momentum10.5.2 The 3-body problem10.6 Variational foundation of fluid motion
11. Analytic mechanics
11.1 Elastic string vibrations11.2 The elastic membrane11.2.1 Circular membrane vibrations11.2.2 Non-zero boundary conditions11.3 Bending of a beam under its own weight11.3.1 Transverse vibration of beam11.4 Buckling of a beam under axial load11.4.1 Axial vibration of a beam11.5 Simultaneous axial and transversal loading of beam11.6 Heat diffusion in a beam11.6.1 Dimensionless heat equation
12. Computational mechanics
12.1 The finite element technique12.1.1 Finite element meshing12.1.2 Shape functions12.1.3 Element matrix generation12.1.4 Element matrix assembly and solution12.2 Three-dimensional elasticity12.3 Mechanical system analysis12.4 Heat conduction12.5 Fluid mechanics
Solutions to selected exercises
Notations
References
Index

Content preview from Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

Multivariate functionals

The subject of this chapter is further expansion of the classes of functions used in variational problems. Specifically, we discuss functionals with multiple functions, functions with multiple variables will be discussed in a variational context. Parametric functions and minimal surface applications will also be presented.

3.1Functionals with several functions

The variational problem of multiple dependent variables is posed as

I (y_{1}, y_{2}, \dots, y_{n}) = \int_{x_{0}}^{x_{1}} f (x, y_{1}, y_{2}, \dots, y_{n}, y_{1}^{'}, y_{2}^{'}, \dots, y_{n}^{'}) d x

with a pair of boundary conditions given for all functions:

y_{i} (x_{0}) = y_{i, 0}

and

y_{i} (x_{1}) = y_{i, 1}

for each i = 1, 2, …, n. The alternative solutions are:

Y_{i} (x) = y_{i} (x) + ϵ_{i} η_{i} (x); i = 1, 2, \dots, n

with all the arbitrary auxiliary functions obeying the conditions: ...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.

Read now

Unlock full access

More than 5,000 organizations count on O’Reilly

O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.

Julian F.

Head of Cybersecurity

I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.

Addison B.

Field Engineer

I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.

Amir M.

Data Platform Tech Lead

I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.

Mark W.

Embedded Software Engineer

Applied Calculus of Variations for Engineers, 2nd Edition

Publisher Resources

ISBN: 9781000764758

Cloud Computing

Data Engineering

Data Science

AI & ML

Programming Languages

Software Architecture

IT/Ops

Security

Design

Business

Soft Skills

Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

by Louis Komzsik

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.