ADVANCES IN COMPUTATIONAL MATERIALS AND STRUCTURES · 2013. 7. 23. · 11.3.2 Space-Discrete Lagrangian Finite Element Formulation / 451 11.4 Hamiltonian Mechanics Framework in the

ADVANCES IN COMPUTATIONALDYNAMICS OF PARTICLES,MATERIALS AND STRUCTURES

ADVANCES IN COMPUTATIONALDYNAMICS OF PARTICLES,MATERIALS AND STRUCTURES

JASON HAR and KUMAR K. TAMMADepartment of Mechanical Engineering, University of Minnesota, Minneapolis, USA

A John Wiley & Sons, Ltd., Publication

© 2012, John Wiley & Sons, Ltd

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Library of Congress Cataloguing-in-Publication Data

Har, Jason.Advances in computational dynamics of particles, materials and structures/

Jason Har, Kumar K. Tamma.p. cm.

Includes bibliographical references and index.ISBN 978-0-470-74980-7 (hardback)

1. Dynamics. 2. Dynamics – Data processing. I. Tamma, Kumar K. II. Title.TA352.H365 2012531′.163 – dc23

2011044208

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-74980-7

Set in 10/12 Times by Laserwords Private Limited, Chennai, India

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To our families, friends and students

CONTENTS

PREFACE xv

ACKNOWLEDGMENTS xxi

ABOUT THE AUTHORS xxiii

1 INTRODUCTION 11.1 Overview / 1

1.1.1 The Mechanics Underlying Computational Dynamics / 21.1.2 The Numerics Underlying Computational Dynamics in Space and Time / 8

1.2 Applications / 13

2 MATHEMATICAL PRELIMINARIES 152.1 Sets and Functions / 15

2.1.1 Sets / 152.1.2 Functions / 17

2.2 Vector Spaces / 182.2.1 Real Vector Spaces / 182.2.2 Linear Dependence and Independence of Vectors / 192.2.3 Euclidean n-Space / 202.2.4 Inner Product Space / 212.2.5 Metric Spaces / 222.2.6 Normed Space / 23

2.3 Matrix Algebra / 242.3.1 Determinant of a Coefficient Matrix / 252.3.2 Matrix Multiplication / 27

viii CONTENTS

2.4 Vector Differential Calculus / 282.4.1 Scalar-Valued Functions of Multivariables / 282.4.2 Vector-Valued Functions of Multivariables / 30

2.5 Vector Integral Calculus / 322.5.1 Green’s Theorem in the Plane / 322.5.2 Gauss’s Theorem / 33

2.6 Mean Value Theorem / 332.6.1 Scalar Function of a Real Variable / 332.6.2 Scalar Function of Multivariables / 342.6.3 Vector Function of Multivariables / 34

2.7 Function Spaces / 342.7.1 Inner Product Space / 352.7.2 Normed Space / 352.7.3 Metric Space / 362.7.4 Lebesgue Space / 362.7.5 Banach Space / 362.7.6 Sobolev Space / 372.7.7 Hilbert Space / 38

2.8 Tensor Analysis / 382.8.1 Tensor Algebra / 392.8.2 Tensor Differential Calculus / 492.8.3 Tensor Integral Calculus / 52Exercises / 53

PART I N-BODY DYNAMICAL SYSTEMS

3 CLASSICAL MECHANICS 573.1 Newtonian Mechanics / 57

3.1.1 Newton’s Laws of Motion / 573.1.2 Newton’s Equations of Motion / 58

3.2 Lagrangian Mechanics / 603.2.1 Constraints / 613.2.2 Lagrangian Form of D’Alembert’s Principle / 683.2.3 Configuration Space / 723.2.4 Generalized Coordinates / 723.2.5 Tangent Bundle / 743.2.6 Lagrange’s Equations of Motion / 763.2.7 Kinetic Energy in Generalized Coordinates / 843.2.8 Lagrange Multiplier Method / 863.2.9 Autonomous Lagrangian Systems / 87

3.3 Hamiltonian Mechanics / 913.3.1 Phase Space / 913.3.2 Canonical Coordinates / 92

CONTENTS ix

3.3.3 Cotangent Bundle / 923.3.4 Legendre Transformation / 933.3.5 Hamilton’s Equations of Motion / 963.3.6 Autonomous Hamiltonian Systems / 993.3.7 Symplectic Manifold / 100Exercises / 103

4 PRINCIPLE OF VIRTUAL WORK 1084.1 Virtual Work in N-Body Dynamical Systems / 1084.2 Vector Formalism: Newtonian Mechanics in N-Body Dynamical Systems / 1144.3 Scalar Formalisms: Lagrangian and Hamiltonian Mechanics in N-Body Dynamical

Systems / 116Exercises / 120

5 HAMILTON’S PRINCIPLE AND HAMILTON’S LAW OF VARYING ACTION 1215.1 Introduction / 1215.2 Variation of the Principal Function / 1225.3 Calculus of Variations / 1255.4 Hamilton’s Principle / 1295.5 Hamilton’s Law of Varying Action / 133

5.5.1 Newtonian Mechanics / 1355.5.2 Lagrangian Mechanics / 1365.5.3 Hamiltonian Mechanics / 137Exercises / 138

6 PRINCIPLE OF BALANCE OF MECHANICAL ENERGY 1416.1 Introduction / 1426.2 Principle of Balance of Mechanical Energy / 1426.3 Total Energy Representations and Framework in the Differential Calculus

Setting / 1446.3.1 Principle of Balance of Mechanical Energy: Conservative System / 1456.3.2 Principle of Balance of Mechanical Energy: Nonconservative System / 1496.3.3 Newtonian Dynamical System: With/Without Constraints / 1516.3.4 Lagrangian Dynamical System: Nonconservative/Conservative Systems -

Descriptive Scalar Function, the Lagrangian / 1526.3.5 Hamiltonian Dynamical System: Nonconservative Systems - Descriptive Scalar

Function, the Hamiltonian / 1546.4 Appendix: Total Energy Representations and Framework in the Variational Calculus

Setting / 1566.4.1 Total Energy Representation of the Equation of Motion via the Lagrangian

Form of D’Alembert’s Principle/Principle of Virtual Work / 1566.4.2 Total Energy Representation of Equation of Motion via Hamilton’s

Principle/Hamilton’s Law of Varying Action / 158Exercises / 162

x CONTENTS

7 EQUIVALENCE OF EQUATIONS 1637.1 Equivalence in the Lagrangian Form of D’Alembert’s Principle/Principle of Virtual

Work / 1637.2 Equivalence in Hamilton’s Principle or Hamilton’s Law of Varying Action / 1657.3 Equivalence in the Principle of Balance of Mechanical Energy / 1667.4 Equivalence Relations Between Governing Equations / 1677.5 Conservation Laws / 1717.6 Noether’s Theorem / 171

Exercises / 172

PART II CONTINUOUS-BODY DYNAMICAL SYSTEMS

8 CONTINUUM MECHANICS 1758.1 Displacements, Strains and Stresses / 175

8.1.1 Configuration Space / 1768.1.2 Riemannian Metrics / 1778.1.3 Infinitesimal Differential Volume / 1788.1.4 Displacements and Strains / 1798.1.5 Stresses / 194

8.2 General Principles / 1978.2.1 Gauss’s Theorem / 1988.2.2 Reynolds Transport Theorem / 1988.2.3 Principle of Conservation of Mass / 2008.2.4 Principle of Balance of Linear Momentum / 2008.2.5 Principle of Balance of Angular Momentum / 2028.2.6 Principle of Balance of Energy / 2038.2.7 Principle of Entropy Inequality / 205

8.3 Constitutive Equations in Elasticity / 2068.3.1 Cauchy Elastic Material / 2068.3.2 Hyperelastic Material / 2078.3.3 Hypoelastic Material / 2138.3.4 Material Frame-Indifference: Objectivity / 2148.3.5 Objective Stress Rates / 218

8.4 Virtual Work and Variational Principles / 2208.4.1 Virtual Work and Potential Energy / 2208.4.2 Principle of Virtual Work / 2238.4.3 Principle of Virtual Power / 2288.4.4 Principle of Stationary Potential Energy / 2288.4.5 Principle of Complementary Virtual Work / 2308.4.6 Principle of Stationary Complementary Energy / 2328.4.7 Hu-Washizu Variational Principle / 233

CONTENTS xi

8.4.8 Hellinger-Reissner Variational Principle / 2358.5 Direct Variational Methods for Two-Point Boundary-Value Problems / 237

8.5.1 Rayleigh-Ritz Method / 2378.5.2 Bubnov-Galerkin Weighted Residual Method / 2448.5.3 Modified Bubnov-Galerkin Weighted-Residual Method / 2478.5.4 Equivalence of the Ritz and the Galerkin Methods / 252Exercises / 253

9 PRINCIPLE OF VIRTUAL WORK: FINITE ELEMENTS AND SOLID/STRUCTURAL MECHANICS 2679.1 Introduction / 267

9.1.1 Vector Formalism: Cauchy’s Equations of Motion, Principle of Virtual Work,and Finite Element Formulations in Continuous-Body Dynamical Systems / 267

9.2 Finite Element Library / 3019.2.1 One-Dimensional Continuum: Axial Bar Element / 3019.2.2 Two-Dimensional Continuum: Triangular Element / 3079.2.3 Two-Dimensional Continuum: Quadrilateral Element / 3139.2.4 Three-Dimensional Continuum: Tetrahedral Element / 3209.2.5 Three-Dimensional Continuum: Hexahedral Element / 3259.2.6 Structural Member: Euler-Bernoulli Beam Element / 3299.2.7 Structural Member: Timoshenko Beam Element / 3339.2.8 Structural Member: Kirchhoff-Love Plate Element / 3369.2.9 Structural Member: Reissner-Mindlin Plate Element / 339

9.3 Nonlinear Finite Element Formulations / 3439.3.1 Total Lagrangian Formulation / 3449.3.2 Updated Lagrangian Formulation / 347

9.4 Scalar Formalisms: Lagrangian and Hamiltonian Mechanics and Finite Element For-mulations in Continuous-Body Dynamical Systems / 350Exercises / 353

10 HAMILTON’S PRINCIPLE AND HAMILTON’S LAW OF VARYING ACTION:FINITE ELEMENTS AND SOLID/STRUCTURAL MECHANICS 36410.1 Introduction / 36410.2 Hamilton’s Principle and Hamilton’s Law of Varying Action

in Elastodynamics / 36510.3 Lagrangian Mechanics Framework and Finite Element Formulations / 370

10.3.1 Lagrangian Density Equations of Motion / 37110.3.2 Space-Discrete Lagrangian Finite Element Formulation / 374

10.4 Hamiltonian Mechanics Framework and Finite Element Formulations / 40010.4.1 Hamiltonian Density Equations of Motion / 40010.4.2 Space-Discrete Hamiltonian Finite Element Formulation / 403Exercises / 425

xii CONTENTS

11 PRINCIPLE OF BALANCE OF MECHANICAL ENERGY: FINITE ELEMENTS ANDSOLID/STRUCTURAL MECHANICS 42611.1 Introduction / 42711.2 Total Energy Representations and Framework in the Differential Calculus

Setting and Finite Element Formulations / 42911.2.1 Principle of Balance of Mechanical Energy/Theorem of Power Expended:

Nonconservative System / 43011.2.2 Principle of Balance of Mechanical Energy: Conservative System and Total

Energy Density Equations of Motion / 43311.2.3 Space-Discrete Total Energy Finite Element Formulation / 436

11.3 Lagrangian Mechanics Framework in the Differential Calculus Setting and FiniteElement Formulations / 44911.3.1 Lagrangian Density Equations of Motion / 44911.3.2 Space-Discrete Lagrangian Finite Element Formulation / 451

11.4 Hamiltonian Mechanics Framework in the Differential Calculus Setting and FiniteElement Formulations / 45411.4.1 Hamiltonian Density Equations of Motion / 45411.4.2 Space-Discrete Hamiltonian Finite Element Formulation / 456

11.5 Appendix: Total Energy Representations and Framework in the Variational CalculusSetting and Finite Element Formulations / 45811.5.1 Infinite Dimensional Total Energy Structure / 45911.5.2 Total Energy Density Representation of the Equation of Motion / 45911.5.3 Space-Discrete Total Energy Finite Element Formulation / 462Exercises / 474

12 EQUIVALENCE OF EQUATIONS 47512.1 Equivalence in the Principle of Virtual Work in Dynamics / 47512.2 Equivalence in Hamilton’s Principle or Hamilton’s Law of Varying Action / 47812.3 Equivalence in the Principle of Balance of Mechanical Energy / 48212.4 Equivalence of Strong and Weak Forms for Initial Boundary-Value Problems / 48312.5 Equivalence of the Semi-Discrete Finite Element Equations of Motion / 48712.6 Equivalence of Finite Element Formulations / 48812.7 Conservation Laws / 490

Exercises / 490

PART III THE TIME DIMENSION

13 TIME DISCRETIZATION OF EQUATIONS OF MOTION: OVERVIEW ANDCONVENTIONAL PRACTICES 49513.1 Introduction / 49513.2 Single-Step Methods for First-Order Ordinary Differential Equations / 500

13.2.1 Forward Euler Method / 501

CONTENTS xiii

13.2.2 Backward Euler Method / 50113.2.3 Generalized Trapezoidal Method / 50213.2.4 Midpoint Method / 50213.2.5 Runge-Kutta Method / 50313.2.6 Generalized Trapezoidal Family for a Vector-Valued Function / 504

13.3 Linear Multistep Methods / 50513.3.1 Central Difference Method / 50513.3.2 Linear Multistep Methods for First-Order Ordinary Differential

Equations / 50613.3.3 Linear Multistep Methods for Second-Order Ordinary Differential

Equations / 50713.4 Second-Order Systems and Single Step and/or Equivalent LMS Methods: Brief

Overview of Classical Methods from Historical Perspectives and ChronologicalDevelopments / 50713.4.1 The Houbolt Method [1950] / 50813.4.2 Classical Midpoint Rule Method / 50913.4.3 The Newmark Family of Algorithms [1959] / 51113.4.4 The Wilson-θ Method [1968] / 51313.4.5 The Park Method [1975] / 51413.4.6 The Hilber-Hughes-Taylor-α Method [1977] / 51513.4.7 The SSpj Family of Algorithms [1977] / 51613.4.8 The Wood-Bosak-Zienkiewicz Method [1981] / 52013.4.9 Velocity Based Scheme [1988] / 52113.4.10 The Three Parameters Optimal Schemes (χ-Schemes) [1988]

(and the identical Generalized-α Method) / 52213.4.11 Optimal U0-V0 Algorithm: The Optimal Algorithm With Controllable

Numerical Dissipation Within the Class of LMS Methods in the Senseof the Single-Field Form [2004] / 525

13.5 Symplectic-Momentum Conservation and Variational Time Integrators / 52713.5.1 Discrete Euler-Lagrange Equations / 52713.5.2 Discrete Legendre Transformation / 53013.5.3 Symplecticness of Variational Time Integrators / 53213.5.4 Discrete Noether’s Theorem / 534

13.6 Energy-Momentum Conservation and Time Integration Algorithms / 53613.6.1 Energy-Momentum Conserving Scheme for Discrete Systems / 53613.6.2 Energy-Momentum Conserving Algorithms for N-Body Systems / 53713.6.3 Energy-Momentum Conserving Algorithms for Continuum

Elastodynamics / 54313.6.4 Extension of Energy-Momentum Conserving Algorithms for a General

Hyperelastic Material Model / 550

14 TIME DISCRETIZATION OF EQUATIONS OF MOTION: RECENT ADVANCES 55314.1 Introduction / 553

xiv CONTENTS

14.2 Time Discretization and the Total Energy Framework: Linear Dynamic Algorithmsand Designs - Generalized Single Step Single Solve [GSSSS] Unified FrameworkEncompassing LMS Methods / 55514.2.1 GSSSS Framework Encompassing LMS Methods: Nonconservative

Systems and Linear Dynamics Algorithms and Designs in Two-field/Single-field Form Via the Semi-Discretized Equations of Motion / 558

14.2.2 GSSSS Framework Encompassing LMS Methods: Conservative Systems andLinear Dynamics Algorithms and Designs in Two-field/Single-field FormVia the Discrete Total Energy Framework / 566

14.2.3 Total Energy Framework and Semi-Discretized Equation of Motion forConservative Dynamic Systems / 569

14.3 Time Discretization and the Total Energy Framework: Nonlinear DynamicsAlgorithms and Designs - Generalized Single Step Single Solve [GSSSS]Framework Encompassing LMS Methods / 57814.3.1 Classical/Normalized Time Weighted Residual Methodology / 57914.3.2 Time Discretization and Total Energy Framework: Brief Highlights of the

Two-/Single-Field Form of LMS Methods, and Conserving Algorithms andDesigns - General Hyperelastic Material Models / 582

14.3.3 Numerical Implementation Aspects of Classical Framework in Single-field Form:Internal Force Based Numerically Non-Dissipative and Dissipative Algorithmsand Designs / 606

14.3.4 Numerical Implementation Aspects of Symplectic-Momentum Framework inSingle-field Form: Symplectic-Momentum Based Numerically Non-Dissipativeand Dissipative Algorithms and Designs / 614

14.3.5 Numerical Implementation Aspects of Energy-Momentum Framework inSingle-field Form: Energy-Momentum Based Numerically Non-Dissipative andDissipative Algorithms and Designs / 622

14.4 Time Discretization and Total Energy Framework: N-Body Systems / 63214.4.1 GSSSS Framework Encompassing LMS Methods: Conservative Systems and

Linear Dynamics Algorithms and Designs in Two-field form and Single-fieldForm Via the Total Energy Framework / 632

14.4.2 GSSSS Framework Encompassing LMS Methods: Conservative Systems andNonlinear Dynamics Algorithms and Designs in Two-field Form and Single-fieldForm Via the Total Energy Framework / 635

14.5 Time Discretization and Total Energy Framework: Nonconservative/ConservativeMechanical Systems with Holonomic-Scleronomic Constraints / 64914.5.1 General Formulations / 650Exercises / 662

REFERENCES 669

INDEX 681

PREFACE

This book treats the subject matter dealing with advances in computational dynamics from a unifiedviewpoint and approach, and thereby provides a rigorous treatment and a unique blend of the vari-ous underlying mechanics and the numerical aspects to effectively foster modeling and simulation onmodern computing environments. In the broader sense, the subject matter under the umbrella of com-putational dynamics covers the necessary fundamentals associated with particle dynamics; dynamicsof materials, structures, deformable continuum media and related applications to include structural/elasto-dynamics; multi-body dynamics dealing with rigid and flexible bodies; contact-impact dynamics;and so on. In particular, this book covers the classical (or traditional) practices to more contemporaryaspects which include recent advances dealing with the mathematical, physical, geometrical, as well ascomputational aspects associated with modeling and simulation as related to numerical discretizationin space and/or time. It is designed for engineers, mathematicians, physicists, and students/researchersin allied fields who wish to understand the subject matter with rigor and in a contemporary setting.We intend this book to serve as a multi-semester course at the graduate-level and/or for upper-levelundergraduate students (on selected topics), advanced researchers and scientists, and engineers whoare keenly interested in the fundamental aspects critical to the computational aspects of the dynamicsof particles and rigid bodies, and the computational aspects dealing with structural/elasto-dynamics,continuum mechanics, the finite element method, and time integration schemes for both N-body andcontinuous-body dynamical systems. This book explores both classical practices as well as new avenueswith differing and alternative viewpoints which additionally provide improved physical insight and newcomputational perspectives. With these considerations in mind, we closely embrace the underlying themeand excerpt due to Gauss as highlighted in Degas (1955): ”It is always interesting and instructive toregard the laws of nature from a new and advantageous point of view, so as to solve this or that problemmore simply, or to obtain a more precise presentation”.

We start with the premise, that in the beginning there were these landmark contributions due to Aristo-tle (384 BC-322 BC), Archimedes (287 BC-212 BC), Galileo (1564-1642), Kepler (1571-1630), Huygens(1629-1695), Decartes (1596-1650), and the like, and, then there was this thing of beauty, namely, that dueto Newton (1643-1727) - the famous Newton’s laws of motion . And now there are all these various fieldsor branches of mechanics and physics with various underlying theoretical pinning’s dealing with particledynamics; dynamics of materials, structures, and deformable continuum media and related applications

xvi PREFACE

to include structural/elasto-dynamics; multi-body dynamics dealing with rigid and flexible bodies;contact-impact dynamics; and the like. It is worth noting that the fundamental principles of dynamicshave also been abstracted to various other fields and applications to include the theory of relativity,quantum mechanics, economics, robotics, biology, medical and allied applications such as biomechanics,virtual surgery physics based simulations for training medical residents/physicians, and the like.

Keeping the above considerations in perspective, we present an overview of not only the classicaldevelopments and the current state-of-the-art, but we also provide new and recent advances dealingwith computational aspects related to the dynamics of particles, materials, and structures. In this book,we first highlight the big picture with consistent developments from differing viewpoints not only toderive the governing equations of motion for N-body or continuous-body dynamical systems for a wideclass of engineering applications, but also to subsequently enable the discretization in space/time fornumerical computations. In particular, we present our viewpoint of the evolution of a variety of numericaldevelopments in the fields encompassing computational dynamics ranging from classical practices tomore new and recent advances. Under the umbrella of computational dynamics , at the outset it should beclearly noted that this book is intended to provide a sound and fundamental background on the varioustheoretical and computational aspects; and we classify the evolution of the various related developmentsvia two principal themes, namely, the mechanics underlying computational dynamics and the associatednumerics underlying computational dynamics . Only in selective instances, certain theoretical bases andrelated considerations dealing with various aspects of classical mechanics have been carefully excerptedand interpreted from several renowned books such as Mach (1907), Pars (1965), Greenwood (1977),Rosenberg (1977), Arnold (1989), Goldstein (2002), and the like which have been some of the primarysources.

Mechanics Underlying Computational Dynamics: The terminology, namely, the mechanics under-lying computational dynamics , implies the approach and starting point that is employed as the funda-mental axiom via which one can independently derive the governing equations, and the associated strongand/or weak forms that can be readily employed for the associated numerical discretizations. Startingwith the premise that in the beginning the well known Newton’s law of motion for the dynamics ofN-body systems is given, which reflects the statement of the principle of balance of linear momentum,subsequently, using this as a landmark, firstly, the principal relations to various other distinctly differentfundamental principles which are of primary interest here are established. This is worth noting. Likewise,for the dynamics of materials, structures, and deformable continuum medium and related applications,under the premise that the governing equations such as the well known Cauchy equations of motionwhich also reflect the statement of the principle of balance of linear momentum are given, analogousrelations as in N-body systems are also established. After first establishing the principal relations to thevarious fundamental principles, any of the respective principles thenceforth can serve as the standalonestarting point for the subsequent theoretical and computational developments for modeling and simu-lation. In this book, we confine attention primarily to three distinctly different fundamental principleswhich comprise the pyramid of computational dynamics . Of particular interest are the three distinctlydifferent fundamental principles represented as faces or planes which comprise the pyramid of com-putational dynamics (see Figure 1), namely: 1) the Principle of Virtual work , 2) Hamilton’s Principle,or alternatively, Hamilton’s Law of Varying Action (which is not a variational principle), and 3) thePrinciple of Balance of Mechanical Energy . Each fundamental principle is particularly selected suchthat it can independently enable the theoretical and computational developments associated with andleading to the strong and/or weak forms and the corresponding numerical discretizations in space/timefor applications to computational dynamics. That is, each of the above fundamental principles does notnecessarily rely upon the others. However, the pros and cons, limitations of each fundamental principle,and the conditions under which equivalences of the respective formulations amongst the three funda-mental principles can be drawn need to be carefully understood to avoid misinterpretation. By no means,we claim that these are the only representations for the classification as various other explanations are

PREFACE xvii

Figure 1. Pyramid of computational dynamics

also plausible and could be included. Consequently, the present pyramidal structure classification couldentertain other faces or planes. However, we confine attention only to the present three fundamentalprinciples with the clear understanding of the restrictions inherent within each fundamental principle.

Numerics Underlying Computational Dynamics: Subsequently, we also describe the numericsunderlying computational dynamics which deals with both classical (or traditional) practices and newavenues for conducting space/time discretizations to find numerical solutions useful for modeling andsimulation. The terminology, numerics underlying computational dynamics , refers to the approach andthe starting point that is employed by which we address the numerical treatments as related to spatialdiscretizations in the space domain and temporal discretizations in the time domain. It deals with thenumerical aspects and discretization approaches in space and/or time which are necessary ingredients formodeling and simulation. Stemming independently from each of the respective fundamental principlescomprising the pyramid of computational dynamics, we describe the various computational developmentsfor the dynamics of N-body systems, and the dynamics of materials, structures, deformable continuummedia and related applications. Both classical practices that are customarily followed, as well as otheralternative avenues which provide new and different perspectives and/or improved physical insight forthe modeling and simulation of computational dynamics applications are described. A unified viewpointis the end result regardless of which fundamental principle serves as the starting point; and the restrictionsand/or limitations associated with each of the respective fundamental principles need to be carefullyunderstood Tamma (2012); Tamma et al. (2011) (DOI10.1007/s11831-011 9060-y).

xviii PREFACE

Outline of this Book: The outline of this book is as follows. Chapter 1 presents anintroduction to and an overview of the big picture and our viewpoint of the various theoreticaland numerical aspects dealing with computational dynamics . Along the themes, namely, the mechanicsunderlying computational dynamics and under the umbrella of the pyramid of computational dynamics ,and the associated numerics underlying computational dynamics , in this book, we focus attention uponthe three fundamental principles comprising the pyramid structure classification, namely: 1) the Princi-ple of Virtual Work , 2) Hamilton’s Principle, or alternatively, Hamilton’s Law of Varying Action (whichis not a variational principle), and 3) the Principle of Balance of Mechanical Energy . Each of the abovefundamental principles has a wide range of applicability, and can independently describe the theoreticaland numerical developments associated with and leading to the strong and/or weak forms and the cor-responding numerical discretizations in space/time for applications to computational dynamics. Chapter2 provides the basic mathematical background materials necessary for studying classical mechanics,continuum mechanics, finite element theories, and time integration schemes for integrating the equationsof motion. Throughout the book, it is very important to have a fundamental grasp of the concepts of setsand functions, and the meaning of the related notations. Vector spaces with numeric entries as well asfunctions are addressed in this chapter. In the discussion on tensor analysis, we use not only Cartesiantensors but also general tensors, which are crucial for understanding nonlinear continuum mechanics andfinite deformation theories for deformable bodies. The book is divided into three parts. Consequently,under the umbrella of the pyramid of computational dynamics , we devote a separate chapter in bothN-Body Systems (Part 1) and Continuous-Body Systems (Part 2) to each of these respective principleswhich independently serve as a starting point for conducting the theoretical and numerical developmentsassociated with and leading to the strong and/or weak forms and the corresponding numerical discretiza-tions in space and/or time. An overview of conventional practices, and in addition, recent advancesdealing with a wide variety of Time Discretization (Part 3) approaches and related time integrationaspects necessary for appropriately integrating the dynamic equations of motion are finally highlighted.

Part 1: N-Body Systems With the above considerations in mind, in Part 1 which deals with N-body Systems, Chapter 3 covers classical mechanics including Newtonian, Lagrangian and Hamiltonianmechanics. In Chapter 4, after first establishing the relation between Newton’s second law and theprinciple of virtual work (which is a restatement of the Lagrangian form of D’Alembert’s principle), wedirectly show the subsequent theoretical and numerical developments starting from this principle. TheLagrangian form of D’Alembert’s principle (or equivalently, the principle of virtual work in dynamics)is the key principle leading to analytical mechanics and descriptive scalar function formalism, in contrastto the Newtonian mechanics framework and vector formalism. Alternatively, Chapter 5 describes bothHamilton’s principle and Hamilton’s law of varying action for N-body dynamical systems. We drawattention in the book to the fact that Hamilton’s law of varying action is equivalent to the integral form ofthe principle of virtual work. Consequently, it is a descriptive scalar function representation of the prin-ciple of virtual work, which naturally contains the weighted residual form in time for N-body dynamicalsystems. In contrast, Chapter 6 describes the principle of balance of mechanical energy as the startingpoint, and the corresponding formulations associated with the Total Energy representation of the equationof motion and framework in the differential calculus setting which is valid for holonomic-scleronomicsystems with a new, measurable, and built-in descriptive scalar function, namely, the Total Energy (andin addition, the variational calculus setting which is valid for holonomic systems is also highlighted inthe Appendix). As a descriptive scalar function analogous to the Lagrangian and the Hamiltonian, theTotal Energy defined on the velocity phase space is yet another alternative and it offers good physi-cal insight and computationally attractive features. There exist various subject areas in mechanics andphysics where it is desirable to have a direct measurable descriptive scalar function such as the TotalEnergy . The related developments also readily enable the theoretical and numerical formulations forcomputational dynamics just as those obtained from the other two fundamental principles. Next, Chapter7 describes equivalence relations between governing equations for N-body dynamical systems subject

PREFACE xix

to holonomic constraints within the three frameworks, namely, the Lagrangian, Hamiltonian and TotalEnergy frameworks. Noether’s Theorem for N-body dynamical systems, and the invariant properties,namely, the conservation of Linear Momentum, Angular Momentum and Total Energy of the descriptivescalar functions, such as the Lagrangian, Hamiltonian and Total Energy are also highlighted.

Part 2: Continuous-Body Systems Part 2 focuses upon Continuous-body Systems, and the contin-uum mechanics aspects associated with deformations, strains, and stresses in solid/structural applications.In Chapter 8, we start with basic continuum mechanics materials necessary for developing finite elementformulations. Chapter 8 describes displacements, strains, and stresses with general tensors. We then dis-cuss five fundamental principles dealing with thermo-mechanical motion which continuous bodies mustobey; these include, the principle of conservation of mass, the principle of balance of linear momentum,the principle of balance of angular momentum, the principle of balance of energy, and the principle ofentropy inequality. Chapter 8 also includes constitutive equations in elasticity, fundamentals of virtualwork and variational principles, and direct variational methods for two-point boundary-value problemssuch as the Rayleigh-Ritz method, the Bubnov-Galerkin weighted residual method, and the modifiedBubnov-Galerkin weighted residual method. As in N-body systems described in Part 1, we next devotea separate chapter dealing with continuous-body systems to each of the three fundamental principlescomprising the pyramid of dynamics which independently serve as the starting point for developing therelated theoretical and numerical formulations. In this regard, Chapter 9 comprehensively deals withthe first of the three principles outlined earlier, namely, the principle of virtual work in dynamics; andconsequently, describes conventional finite element formulations and vector formalism for continuous-body dynamical systems. We additionally describe a variety of structural members including axial bar,rotating circular bar, Euler-Bernoulli beam, Timoshenko beam, Kirchhoff-Love thin plate, and Reissner-Mindlin plate. The weak forms for continuum and structural members are derived from the weightedresidual form. With regards to structural members, we first set up a free-body diagram and count onD’Alembert’s principle to obtain the governing equations of motion. Then, we establish the weightedresidual statement to derive the weak form by performing integration by parts and imposing naturalboundary conditions. Finally, the resulting weak form is spatially discretized by using appropriate trialand test functions. In addition to the finite element formulations, we additionally describe a variety offinite elements including axial bar element, plane stress/strain two-dimensional triangular and quadri-lateral elements, three-dimensional tetrahedral and hexahedral brick elements, Euler-Bernoulli beamelement, Timoshenko beam element, Kirchhoff-Love plate element and Reissner-Mindlin plate element.Lastly, we highlight nonlinear finite element formulations including total and updated Lagrangian for-mulations. Scalar formalisms with respect to the Lagrangian and the Hamiltonian are briefly referenced.In Chapter 10, in contrast to the traditional practices described in Chapter 9, we present finite elementformulations using descriptive scalar functions via Hamilton’s Principle or Hamilton’s Law of VaryingAction as the starting point which also yield the same and/or equivalent finite element representationsfrom another viewpoint. As yet another alternative with several computationally attractive features andgood physical insight, in Chapter 11 we describe other related developments via the Total Energy rep-resentations and framework for developing the finite element formulations using the various descriptivescalar functions. This is via the theorem of power expended, and consequently the principle of balanceof mechanical energy with differential calculus setting valid for holonomic-scleronomic systems. In theAppendix, we briefly also highlight the variational calculus setting in the context of the Total Energyrepresentations and framework which is valid for holonomic systems. Chapter 12 discusses the equiva-lences between the strong forms and also between the weak forms which are respectively obtained viaeach of the three distinctly different fundamental principles, namely, the principle of virtual work, Hamil-ton’s principle or equivalently, Hamilton’s law of varying action, and the theorem of power expendedand consequently the principle of balance of mechanical energy. We also present a brief discussionon Noether’s Theorem for continuous-body dynamical systems, wherein after the spatial discretization

xx PREFACE

they lead to finite dimensional systems analogous to the discussion highlighted in Chapter 7 for N-bodysystems.

Part 3: Time Discretization Finally, Part 3 is devoted to the Time Dimension and the numericalaspects that are necessary for properly dealing with the time integration of the equations of motionin both single-field and two-field forms of representation Tamma (2012); Tamma et al. (2011 (DOI10.1007/s11831-011-9060-y). For the time discretization, an overview of the big picture and specificguidelines for developing algorithms by design that meet targeted objectives are provided and discussed.In Chapter 13 we present the following: (i) We first show starting from the standard representation ofthe linear semi-discretized equations of motion, the various classical and chronological developmentsin time integration of linear dynamical systems from historical perspectives that appear in the open lit-erature over the past fifty years or so, (ii) Next, we highlight variational integrators stemming from theso-called Discrete Euler-Lagrange representations that inherit features which are symplectic-momentumconserving, and (iii) Following this, we highlight the so-called energy-momentum conserving/dissipatingalgorithm designs for finite dimensional systems following the original methods of development (classi-cal practices) through enforcing energy constraints. Lastly, in Chapter 14, in contrast to all the previouslymentioned classical and/or traditional practices described in Chapter 13, we focus special attention uponand highlight the more recent developments directly emanating from the new Total Energy frameworkand representations as a starting point (unlike traditional practices) in conjunction with a generalizedtime weighted residual approach. In particular, we provide new perspectives, a unified viewpoint, andin addition, the underlying theoretical basis on how to properly provide appropriate extensions of theparent linear dynamics algorithm designs to nonlinear dynamics applications for developing practicalalgorithms by design useful for integrating the equations of motion; and the associated computation-ally attractive features are that the developments are based upon symplectic-momentum conservation orenergy-momentum conservation aspects, respectively. These latter developments via the Total Energyrepresentations and framework, and the generalized time weighted residual approach also cover most ofthe developments that have been previously derived from various other classical viewpoints as mentionedin (i), (ii), and (iii) previously in Chapter 13. In summary, for both single-field and two-field forms ofrepresentation, we first describe linear dynamics algorithms by design for integrating the equations ofmotion, and we then provide the necessary theoretical basis for proper extensions to nonlinear dynamicsalgorithms by design. The overall developments are generally applicable to a wide variety of applicationsencompassing linear and nonlinear structural/elasto-dynamics applications in continuous-body dynam-ics, N-body systems, and conservative/nonconservative mechanical systems with holonomic-scleronomicconstraints such as those encountered in multi-body dynamics applications.

Jason Har and Kumar K. Tamma

ACKNOWLEDGMENTS

Professor Kumar K. Tamma is particularly grateful to Dr. Jason Har for his steadfast commitment toembrace the original ideas and concepts that are put forth in this book, learn, and contribute duringthe five and a half year period he served as a post-doctoral associate and research assistant professorunder Professor Tamma’s supervision in the Department of Mechanical Engineering at the Universityof Minnesota. Special thanks are due to Mr. Masao Shimada, graduate Ph.D research student in theDepartment of Mechanical Engineering at the University of Minnesota and working under the supervisionof Professor Kumar K. Tamma, for his valuable technical comments and contributions; in particular onthe time integration aspects.

ABOUT THE AUTHORS

Professor Kumar K. Tamma is a highly recognized researcher and distinguished scholar, and is Profes-sor in the Department of Mechanical Engineering, College of Science and Engineering, at the Universityof Minnesota, Minneapolis, Minnesota. Professor Tamma has published over 200 research papers inarchival journals and book chapters, and over 300 research papers in refereed conference proceed-ings/abstracts. Professor Tamma’s primary areas of research encompass: Computational mechanics withemphasis on multi-scale and multi-physics aspects in space and time, and on design and developmentof novel numerical methods and computational algorithms by design for the modeling and simula-tion of time dependent problems and High Performance Computing applications; multi-disciplinarycomputational fluid-thermal-structural interactions; structural dynamics and large deformation and largestrain contact-impact-penetration-damage; multi-body dynamics of rigid and flexible bodies; computa-tional aspects of macroscale/microscale/nanoscale heat transfer; advanced and lightweight compositesand multifunctional materials manufacturing processes, and solidification. Professor Tamma serves onthe editorial boards for over 15 national and international journals and is the co-editor-in-chief of anonline journal. Professor Tamma is the recipient of numerous research awards including the GeorgeTaylor Research Award for significant and exceptional contributions to research at the University ofMinnesota. Professor Tamma is also the recipient of numerous Outstanding Teacher of the Year andother national and university related awards. Professor Tamma has presented several Plenary/Semi-Plenary/Keynote lectures and various invited lectures in national/international conferences, and acrossvarious government and industrial agencies, and academic institutions. Professor Tamma is a fellowof various related societies in his field, and is also listed in various Who’s Who of organizations andprofessionals.

Dr. Jason Har is a Senior Software Developer with ANSYS, Inc., in Canonsburg, Pennsylvania.Dr. Har worked under the supervision of Professor Kumar K. Tamma at the University of Minnesotain Minneapolis, Minnesota, where Dr. Har learned and embraced the original ideas and concepts beingpursued by Professor Tamma that are put forth in this textbook; and contributed to the various devel-opments for a period of five and a half years as a post doctoral associate and as a research assistant

xxiv ABOUT THE AUTHORS

professor in the Department of Mechanical Engineering at the University of Minnesota. Dr. Jason Harhas extensive industrial experience in finite element technology of structures and structural compo-nents, contact-impact, and parallel computations for over 15 years, and worked at the Korea Institute ofAerospace Technology where Dr. Har also served as Managing Research Director. Dr. Har has presentedvarious invited and special lectures at various organizations and national/international conferences.

a

CHAPTER ONE

INTRODUCTION

The present book encompasses classical (or traditional practices) as well as advances in computationaldynamics for computer modeling and simulation of applications in science and engineering. The high-lights of this book are outlined in Chapter 1. The targeted objectives are towards a wide variety of scienceand engineering problems in particle dynamics; dynamics of materials, structures and deformable con-tinuum media; and related applications which fall under this class of applications. We first introduce inthis book the big picture and a unified viewpoint, and the various approaches which follow for the mod-eling and simulation in the broad field encompassing computational dynamics . In the broader sense, inthis book the subject matter under the umbrella of computational dynamics covers the necessary funda-mentals associated with particle dynamics; dynamics of materials and deformable continuum media andrelated applications to include structural/elasto-dynamics; multi-body dynamics dealing with rigid andflexible bodies; contact-impact dynamics; and so on. We classify the evolution of the various relateddevelopments under the umbrella of computational dynamics via two principal themes, namely, themechanics underlying computational dynamics and the numerics underlying computational dynamics .

1.1 OVERVIEW

An overview of the “big picture” follows next. With regards to the mechanics underlying computationaldynamics , we start with the premise that in the beginning, the well known Newton’s law of motion forN-body systems is given, which reflects the statement of the principle of balance of linear momentum.Subsequently using this as a landmark, firstly, the principal relations to three distinctly different funda-mental principles, that comprise the pyramid of computational dynamics , and are of primary interest hereare established. Likewise, for the dynamics of materials, structures and deformable continuum mediaand related applications, under the premise that the governing equations such as the Cauchy’s equationsof motion which reflect the statement of the principle of balance of linear momentum are given, analo-gous developments are also established. Once the principal relations to the three fundamental principlesare established, any of the respective principles can thenceforth serve as the standalone starting pointfor the subsequent theoretical and numerical developments because of their wide range of applicability.The overall developments provide a fundamental understanding and improved insight into the math-ematical equations governing the dynamic motion for N-body and continuous body systems, and theconsequent numerical discretization in space and/or time. Stemming from the three distinctly different

Advances in Computational Dynamics of Particles, Materials and Structures, First Edition. Jason Har and Kumar K. Tamma.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

2 ADVANCES IN COMPUTATIONAL DYNAMICS OF PARTICLES, MATERIALS AND STRUCTURES

fundamental principles, we present recent advances in both vector and scalar formalisms for N-bodydynamical systems and also continuous-body dynamical systems with focus upon the numerical aspectsrelated to space/time discretization. The three distinctly different fundamental principles which comprisethe pyramid of computational dynamics are the following: the Principle of Virtual Work in Dynamics ,Hamilton’s Principle and as an alternative (due to inconsistencies associated with Hamilton’s principle),Hamilton’s Law of Varying Action , and the Principle of Balance of Mechanical Energy . Essentially, theaforementioned three fundamental principles have been particularly highlighted and selected as each ofthese principles can be independently employed to derive the governing equations of motion for N-bodydynamical systems, and the strong and weak forms for continuous-body dynamical systems. However,of importance and noteworthy are the various formalisms and the different ways by which one candescribe the theoretical and computational developments; and there exist fundamental differences in thethree distinctly different fundamental principles and their underlying axioms.

Customarily in the literature, the equations of motion, which govern the mechanical behavior ofN-body or continuous-body dynamical systems for a wide class of engineering applications, have beenrepresented by vectorial quantities in the Newtonian mechanics framework (which is referred to in thisbook as the vector formalism). Alternatively, in the Lagrangian or Hamiltonian mechanics framework(which is referred to in this book as the scalar formalism), they have been described by generalizedor canonical coordinates with descriptive scalar functions such as the Lagrangian or the Hamiltonian;this is mostly in the sense of applications to N-body dynamical systems (Greenwood 1977; Pars 1965).This has been the traditional paradigm. It is a matter of convenience and preferred choice of theanalyst in the particular selection of either vector or scalar formalism, and the corresponding framework.Although it is not customary in the classical mechanics setting, other alternative descriptive scalarfunctions exist and can also be employed. The significance and importance of one such descriptive scalarfunction which is built-in and directly measurable, namely, the Total Energy, is additionally describedin this book under the umbrella of the Total Energy representation of the equation of motion and theassociated framework . There exist various subject areas in mechanics and physics where it is desirableto have a direct measurable descriptive scalar function such as the Total Energy. It provides a new anddifferent perspective with good physical insight and computationally attractive and convenient featuresin contrast to the classical mechanics setting. The end result is that any of the three previously mentionedfundamental principles can independently be employed to derive the governing equations of motion forN-body dynamical systems, and the strong and weak forms for continuous-body dynamical systems.Also, both the vector and scalar formalisms indeed can be shown to be identical and/or equivalencescan be drawn. Furthermore, each respective framework has its own pros and cons which need to becarefully understood in developing the numerical discretizations in space and/or time. In summary,we describe both classical practices that are customarily followed and new avenues for conductingspace/time discretizations to find numerical solutions.

Under the umbrella of computational dynamics , this book provides a fundamentally sound backgroundon the various theoretical and computational aspects. The two principal themes, namely, the mechanicsunderlying computational dynamics and the associated numerics underlying computational dynamicsare highlighted next. Although in the following the context is in the sense of N-body systems (and isdescribed in detail in Part 1), the corresponding analogy can be equally drawn for Continuous-bodydynamical systems (and is described in detail in Part 2); while the Time Discretization of the equationsof motion is covered in Part 3 of this book.

1.1.1 The Mechanics Underlying Computational Dynamics

In general, classical mechanics is classified into three branches: Newtonian, Lagrangian, and Hamiltonianmechanics. It is believed that the distinction between Newtonian, Lagrangian, and Hamiltonian mechan-ics emanates from the notion of space (Arnold 1989). Alternative descriptive scalar functions exist,

INTRODUCTION 3

but are not the tradition. However, in this book, the significance and importance of one such descrip-tive scalar function which is built-in and directly measurable, namely, the Total Energy, is additionallydescribed under the umbrella of the Total Energy representation of the equation of motion and thecorresponding framework. Some brief highlights of the various frameworks follow next.

Newtonian Mechanics and Framework Newton’s Philosphiae Naturalis Principia Mathematica(Newton, 1687) is based upon Euclid’s axiomatic Elements of Geometry, which comprises of a multitudeof definitions, axioms, theorems, and geometrical constructions in the course of the developments. Theunderlying theory as related to the so called particle mechanics which is often referred to as particledynamics in the classical sense of the Newtonian mechanics setting has been, and is related to afundamentally sound premise which is the main starting point, namely, Newton’s second law. It reflectsthe statement of the principle of balance of linear momentum and is quite widely employed in dynamics.This law relates force to mass and acceleration wherein the velocities are continuous (unlike those specialsituations which are not Newtonian in the strict sense, such as when the velocities have isolated finitediscontinuities resulting in the relation impulse equals change of linear momentum). Accepting therestrictions of Newton’s law (for example, it is not applicable to a broad range of physical phenomena),and within these confines the underlying principles are, however, indeed those associated with theconcepts involving the setting of vectorial dynamics; the basic formalism is with vector representations.

In three-dimensional Euclidean space, Newtonian mechanics requires the existence of an inertialframe of reference, where Newton’s laws of motion hold. The inertial frame of reference is a non-rotating and non-accelerating frame of reference (Gron and Hervik 2007; McComb 1999). Then, as adirect consequence of the statement of the principle of balance of linear momentum, the Newtoniandynamical system (Newton’s equation of motion) is described by physical quantities, which are rep-resented by vectors such as position, velocity, acceleration and force in three-dimensional Euclideanspace. This is referred to as vector formalism in this book. For cases when the Newtonian dynam-ical system is subjected to constraints in terms of Cartesian coordinate variables (which are usuallyused but not required to describe the motion of the dynamical system), these variables are frequentlyemployed to impose constraints which limit the motion of the system in three-dimensional Euclideanspace. The Newtonian dynamical system requires the description of field vectors in Cartesian coordi-nates, suffers from the presence of k-number of constraints leading an N-body dynamical system with3N − k degrees of freedom (Goldstein 2002), and are given in an inertial reference frame. Note thatthe Newtonian dynamical system involves 3N-number of Cartesian variables. Consequently, in Newto-nian mechanics the governing equation of motion is of second-order in time (single-field form with theposition as the dependent variable), represented in terms of Cartesian coordinates, and is subjected toconstraint functions with Cartesian coordinate variables. Strictly speaking, this is referred to as the New-tonian mechanics framework. However, loosely speaking, we shall use the term Newtonian mechanicsframework in this book for the procedure which treats the mathematical developments with vectors suchas position, velocity, acceleration and force in the Cartesian coordinate system.

Subsequent major milestones were then followed by Bernoulli (1700–1782), D’Alembert(1717–1782), Euler (1707–1783), Lagrange (1736–1813), and Hamilton (1805–1865), and so onincluding Riemann (1822–1866), Lie (1842–1899), Poincare (1854–1912), Einstein (1879–1955), andNoether (1882–1935).

Lagrangian Mechanics and Framework In contrast to the Newtonian mechanics setting and vectorformalism, and under the umbrella of analytical dynamics, this field, whose foundations were laid downby Euler as early as 1783, was put into play by Lagrange (Mechanique Analytique (Lagrange 1788),building upon the work of D’Alembert in 1743). Lagrange makes the following claim which is ourinterpretation of his original work: I have set forth a theory in mechanics and the science of solutionto such problems via general formulations which are simple and yield all necessary equations for their

4 ADVANCES IN COMPUTATIONAL DYNAMICS OF PARTICLES, MATERIALS AND STRUCTURES

solution. No figures will be found in this theory, and the approaches I outline do not require geometricalconstructions or discussions based on mechanics, but only simple algebraic principles. Those who enjoy(love) analysis will have great pleasure and see this science of mechanics become a new branch, and willbe grateful to me for having extended this field of mechanics . Unlike the previous vector based formalismassociated with Newtonian mechanics, the developments dealing with analytical dynamics invoke insteadthe formalism of scalar representations and are quite often termed as symbolic representations that areframe invariant. They enable a generalized and unified viewpoint for dealing with the equations of motion(and although they are fairly popular for particle dynamics, they have not enjoyed much attention fordeformable continuous bodies; this is especially in the sense of conducting numerical discretizations inspace and time via methods such as the finite element method, and tradition has a lot to do with thisissue). Furthermore, although there is no new physics that is brought forth in contrast to Newtonianmechanics, one cannot trivialize Lagrangian mechanics (it is an alternative route to the same results).Indeed, it also has certain inherent subtleties and restrictions, but it is important to note that in this bookwe are primarily interested in the common, but wide class of problems and applications wherein bothNewtonian and Lagrangian mechanics hold.

Lagrangian mechanics does not need vector quantities requiring the inertial frame of reference.Instead of vector quantities having both magnitude and direction, a descriptive scalar function havingonly magnitude, called the Lagrangian, is required to be defined to describe the motion of the dynamicalsystem (Greenwood 1977; Pars 1965). In addition, the salient feature of Lagrangian mechanics is thenotion and introduction of the concept of generalized coordinates; this makes it possible to eliminateconstraint equations that arise in Cartesian coordinates. Throughout this book, we refer to this procedureas a scalar formalism. The space where the constraint equations associated with the inertial frame ofreference disappear is called configuration space. It should be noted that the Newtonian dynamicalsystem is usually defined in the inertial frame of reference, whereas the autonomous Lagrangian isdefined on the velocity phase space (tangent bundle) and the Lagrangian dynamical system (Lagrange’sequation of motion) is given in the configuration space, the size of which is ndof. Note that the numberof degrees of freedom is ndof = 3N − k and the configuration space belongs to Euclidean ndof-space.Consequently, in Lagrangian mechanics, the governing equations of motion (Lagrange’s equations ofmotion), which are represented in terms of generalized coordinates and generalized velocities, are notsubjected to constraint functions with Cartesian coordinate variables. The representation of the equationsof motion is also of second-order in time (single-field form with position as the dependent variable) butinvolves a descriptive scalar function, namely, the Lagrangian. Strictly speaking, this is referred to asthe Lagrangian mechanics framework in this book. However, loosely speaking, we shall use the term,the Lagrangian mechanics framework, for the procedure which treats the mathematical developmentswith the descriptive scalar function, namely, the Lagrangian.

Hamiltonian Mechanics and Framework Alternatively, Hamiltonian mechanics (Hamilton 1834a)has introduced the concept of the so-called phase space (co-tangent bundle configuration space) with2ndof-number of canonical variables. Although it does not have any constraints, it inherits insteadthe representation of the equations of motion as a system of first-order in time via a scalar function,namely, the Hamiltonian. By introducing the notion and concepts of canonical coordinates and by meansof the Legendre transformation, the descriptive scalar function, called the Hamiltonian, is defined inHamiltonian mechanics. These canonical coordinates, as an ordered pair, belong to the domain of theHamiltonian, namely, phase space (cotangent bundle), the size of which is 2ndof. This procedure isalso referred to as a scalar formalism in this book. One of the most important aspects in Hamilton’smechanics is the fact that the Hamiltonian dynamical system (Hamilton’s equations of motion) is asystem of first-order differential equations in time unlike the Newtonian or the Lagrangian system whichinvolves a system of second-order differential equations in time. Again, although it also has not broughtforth any new physics just as in Lagrangian mechanics in contrast to that of Newtonian mechanics, this