Dynamics and statics Books

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Book SynopsisIn the tradition of illustrated science bestsellers, like Thing Explainer and harkening back to the classic film The Powers of Ten, this unique, fully-illustrated, four-color book explores and visualizes the concept of scale in our universe.

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Book SynopsisThis volume provides a comprehensive treatment of modern computational mechanics work in particle and continuum dynamics. The coverage encompasses classical Newtonian, Lagrangian, and Hamiltonian mechanics, as well as new and alternate contemporary approaches and their equivalences to address various problems in engineering sciences and physics.Table of ContentsPREFACE xv ACKNOWLEDGMENTS xxi ABOUT THE AUTHORS xxiii 1 INTRODUCTION 11.1 Overview 11.2 Applications 13 2 MATHEMATICAL PRELIMINARIES 152.1 Sets and Functions 152.2 Vector Spaces 182.3 Matrix Algebra 242.4 Vector Differential Calculus 282.5 Vector Integral Calculus 322.6 Mean Value Theorem 332.7 Function Spaces 342.8 Tensor Analysis 38 PART I N-BODY DYNAMICAL SYSTEMS 3 CLASSICAL MECHANICS 573.1 Newtonian Mechanics 573.2 Lagrangian Mechanics 603.3 Hamiltonian Mechanics 91 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 116 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 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 1446.4 Appendix: Total Energy Representations and Framework in the Variational Calculus Setting 156 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 PART II CONTINUOUS-BODY DYNAMICAL SYSTEMS 8 CONTINUUM MECHANICS 1758.1 Displacements, Strains and Stresses 1758.2 General Principles 1978.3 Constitutive Equations in Elasticity 2068.4 Virtual Work and Variational Principles 2208.5 Direct Variational Methods for Two-Point Boundary-Value Problems 237 9 PRINCIPLE OF VIRTUAL WORK: FINITE ELEMENTS AND SOLID/STRUCTURAL MECHANICS 2679.1 Introduction 2679.2 Finite Element Library 3019.3 Nonlinear Finite Element Formulations 3439.4 Scalar Formalisms: Lagrangian and Hamiltonian Mechanics and Finite Element Formulations in Continuous-Body Dynamical Systems 350 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 37010.4 Hamiltonian Mechanics Framework and Finite Element Formulations 400 11 PRINCIPLE OF BALANCE OF MECHANICAL ENERGY: FINITE ELEMENTS AND SOLID/STRUCTURAL MECHANICS 42611.1 Introduction 42711.2 Total Energy Representations and Framework in the Differential Calculus Setting and Finite Element Formulations 42911.3 Lagrangian Mechanics Framework in the Differential Calculus Setting and Finite Element Formulations 44911.4 Hamiltonian Mechanics Framework in the Differential Calculus Setting and Finite Element Formulations 45411.5 Appendix: Total Energy Representations and Framework in the Variational Calculus Setting and Finite Element Formulations 458 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 PART III THE TIME DIMENSION 13 TIME DISCRETIZATION OF EQUATIONS OF MOTION: OVERVIEW AND CONVENTIONAL PRACTICES 49513.1 Introduction 49513.2 Single-Step Methods for First-Order Ordinary Differential Equations 50013.3 Linear Multistep Methods 50513.4 Second-Order Systems and Single Step and/or Equivalent LMS Methods: Brief Overview of Classical Methods from Historical Perspectives and Chronological Developments 50713.5 Symplectic-Momentum Conservation and Variational Time Integrators 52713.6 Energy-Momentum Conservation and Time Integration Algorithms 536 14 TIME DISCRETIZATION OF EQUATIONS OF MOTION: RECENT ADVANCES 55314.1 Introduction 55314.2 Time Discretization and the Total Energy Framework: Linear Dynamic Algorithms and Designs - Generalized Single Step Single Solve [GSSSS] Unified Framework Encompassing LMS Methods 55514.3 Time Discretization and the Total Energy Framework: Nonlinear Dynamics Algorithms and Designs - Generalized Single Step Single Solve [GSSSS] Framework Encompassing LMS Methods 57814.4 Time Discretization and Total Energy Framework: N-Body Systems 63214.5 Time Discretization and Total Energy Framework: Nonconservative/Conservative Mechanical Systems with Holonomic-Scleronomic Constraints 64914.5.1 General Formulations 650Exercises 662 REFERENCES 669 INDEX 681

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Book SynopsisStatistical mechanics is one of the most important areas of physics, and it also has applications to subjects as diverse as economics, social behavior, algorithmic theory, and evolutionary biology. This bokk introduces important developments in classical statistical mechanics, and guides readers to the very threshold of research.Trade Review"Unlike typical textbooks ... [Statistical Mechanics in a Nutshell] presents statistical mechanics as a more general theory with broader applications... A graduate student or researcher who wants to explore the applications of statistical mechanics would be very well served by this book."--Choice "Peliti's Statistical Mechanics in a Nutshell is a fantastic reference for those who know the subject, teach it, or need a quick technical reminder, especially on the topic of phase transitions, which are consistently featured in modern-day discussions... Statistical Mechanics in a Nutshell provides the more general overview, with topics such as the renormalization group method. It includes a good mix of fundamental thermodynamics, phase behaviour, and other key subjects."--Physics TodayTable of ContentsPreface to the English Edition xi Preface xiii Chapter 1: Introduction 1 1.1 The Subject Matter of Statistical Mechanics 1 1.2 Statistical Postulates 3 1.3 An Example: The Ideal Gas 3 1.4 Conclusions 7 Recommended Reading 8 Chapter 2: Thermodynamics 9 2.1 Thermodynamic Systems 9 2.2 Extensive Variables 11 2.3 The Central Problem of Thermodynamics 12 2.4 Entropy 13 2.5 Simple Problems 14 2.6 Heat and Work 18 2.7 The Fundamental Equation 23 2.8 Energy Scheme 24 2.9 Intensive Variables and Thermodynamic Potentials 26 2.10 Free Energy and Maxwell Relations 30 2.11 Gibbs Free Energy and Enthalpy 31 2.12 The Measure of Chemical Potential 33 2.13 The Koenig Born Diagram 35 2.14 Other Thermodynamic Potentials 36 2.15 The Euler and Gibbs-Duhem Equations 37 2.16 Magnetic Systems 39 2.17 Equations of State 40 2.18 Stability 41 2.19 Chemical Reactions 44 2.20 Phase Coexistence 45 2.21 The Clausius-Clapeyron Equation 47 2.22 The Coexistence Curve 48 2.23 Coexistence of Several Phases 49 2.24 The Critical Point 50 2.25 Planar Interfaces 51 Recommended Reading 54 Chapter 3: The Fundamental Postulate 55 3.1 Phase Space 55 3.2 Observables 57 3.3 The Fundamental Postulate: Entropy as Phase-Space Volume 58 3.4 Liouville's Theorem 59 3.5 Quantum States 63 3.6 Systems in Contact 66 3.7 Variational Principle 67 3.8 The Ideal Gas 68 3.9 The Probability Distribution 70 3.10 Maxwell Distribution 71 3.11 The Ising Paramagnet 71 3.12 The Canonical Ensemble 74 3.13 Generalized Ensembles 77 3.14 The p-T Ensemble 80 3.15 The Grand Canonical Ensemble 82 3.16 The Gibbs Formula for the Entropy 84 3.17 Variational Derivation of the Ensembles 86 3.18 Fluctuations of Uncorrelated Particles 87 Recommended Reading 88 Chapter 4: Interaction-Free Systems 89 4.1 Harmonic Oscillators 89 4.2 Photons and Phonons 93 4.3 Boson and Fermion Gases 102 4.4 Einstein Condensation 112 4.5 Adsorption 114 4.6 Internal Degrees of Freedom 116 4.7 Chemical Equilibria in Gases 123 Recommended Reading 124 Chapter 5: Phase Transitions 125 5.1 Liquid-Gas Coexistence and Critical Point 125 5.2 Van der Waals Equation 127 5.3. Other Singularities 129 5.4 Binary Mixtures 130 5.5 Lattice Gas 131 5.6 Symmetry 133 5.7 Symmetry Breaking 134 5.8 The Order Parameter 135 5.9 Peierls Argument 137 5.10 The One-Dimensional Ising Model 140 5.11 Duality 142 5.12 Mean-Field Theory 144 5.13 Variational Principle 147 5.14 Correlation Functions 150 5.15 The Landau Theory 153 5.16 Critical Exponents 156 5.17 The Einstein Theory of Fluctuations 157 5.18 Ginzburg Criterion 160 5.19 Universality and Scaling 161 5.20 Partition Function of the Two-Dimensional Ising Model 165 Recommended Reading 170 Chapter 6: Renormalization Group 173 6.1 Block Transformation 173 6.2 Decimation in the One-Dimensional Ising Model 176 6.3 Two-Dimensional Ising Model 179 6.4 Relevant and Irrelevant Operators 183 6.5 Finite Lattice Method 187 6.6 Renormalization in Fourier Space 189 6.7 Quadratic Anisotropy and Crossover 202 6.8 Critical Crossover 203 6.9 Cubic Anisotrophy 208 6.10 Limit n 209 6.11 Lower and Upper Critical Dimensions 213 Recommended Reading 214 Chapter 7: Classical Fluids 215 7.1 Partition Function for a Classical Fluid 215 7.2 Reduced Densities 219 7.3 Virial Expansion 227 7.4 Perturbation Theory 244 7.5 Liquid Solutions 246 Recommended Reading 249 Chapter 8: Numerical Simulation 251 8.1 Introduction 251 8.2 Molecular Dynamics 253 8.3 Random Sequences 259 8.4 Monte Carlo Method 261 8.5 Umbrella Sampling 272 8.6 Discussion 274 Recommended Reading 275 Chapter 9: Dynamics 277 9.1 Brownian Motion 277 9.2 Fractal Properties of Brownian Trajectories 282 9.3 Smoluchowski Equation 285 9.4 Diffusion Processes and the Fokker-Planck Equation 288 9.5 Correlation Functions 289 9.6 Kubo Formula and Sum Rules 292 9.7 Generalized Brownian Motion 293 9.8 Time Reversal 296 9.9 Response Functions 296 9.10 Fluctuation-Dissipation Theorem 299 9.11 Onsager Reciprocity Relations 301 9.12 Affinities and Fluxes 303 9.13 Variational Principle 306 9.14 An Application 308 Recommended Reading 310 Chapter 10: Complex Systems 311 10.1 Linear Polymers in Solution 312 10.2 Percolation 321 10.3 Disordered Systems 338 Recommended Reading 356 Appendices 357 Appendix A Legendre Transformation 359 A.1 Legendre Transform 359 A.2 Properties of the Legendre Transform 360 A.3 Lagrange Multipliers 361 Appendix B Saddle Point Method 364 B.1 Euler Integrals and the Saddle Point Method 364 B.2 The Euler Gamma Function 366 B.3 Properties of N-Dimensional Space 367 B.4 Integral Representation of the Delta Function 368 Appendix C A Probability Refresher 369 C.1 Events and Probability 369 C.2 Random Variables 369 C.3 Averages and Moments 370 C.4 Conditional Probability: Independence 371 C.5 Generating Function 372 C.6 Central Limit Theorem 372 C.7 Correlations 373 Appendix D Markov Chains 375 D.1 Introduction 375 D.2 Definitions 375 D.3 Spectral Properties 376 D.4 Ergodic Properties 377 D.5 Convergence to Equilibrium 378 Appendix E Fundamental Physical Constants 380 Bibliography 383 Index 389

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Book SynopsisTrade Review"This thought-provoking and unique presentation of the semiclassical approach to quantum physics is by a grandmaster of the subject. All the explanations are original and the illustrations are beautiful. The subject deserves to be better known to researchers in physics and chemistry."—Michael Berry, University of Bristol"This book captures a lifetime of research, achievement, and deep understanding of the semiclassical approach to quantum mechanics. I know of no volume that covers the same eclectic mix of topics, and Heller's insights are invaluable. A heroic undertaking, this book will be a tremendous boon to many research fields."—Kieron Burke, University of California, Irvine"Among the books on quantum mechanics, this one is unique due to the originality of its content, presentation, and interpretation of the results. Heller succeeds in demonstrating remarkable and surprising connections between classical and quantum mechanics, which allows him to explain seemingly complicated quantum-mechanical phenomena in very simple terms. Filling an important gap in the field, this book will be welcome by specialists and nonspecialists alike."—Jiri Vanicek, École Polytechnique Fédérale de Lausanne

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Book SynopsisJohn Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach--known as Aubry-Mather theory--singles out the existence of special orbits and invariant measures of the system, which posseTable of ContentsPreface vii 1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1 1.1 Lagrangian Point of View 1 1.2 Hamiltonian Point of View 4 2 From KAM Theory to Aubry-Mather Theory 8 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori 8 3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 18 3.1 Action-Minimizing Measures and Mather Sets 18 3.2 Mather Measures and Rotation Vectors 24 3.3 Mather's a-and B-Functions 28 3.4 The Symplectic Invariance of Mather Sets 35 3.5 An Example: The Simple Pendulum (Part I) 39 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians 45 4 Action-Minimizing Curves for Tonelli Lagrangians 48 4.1 Global Action-Minimizing Curves: Aubry and Mane Sets 48 4.2 Some Topological and Symplectic Properties of the Aubry and Mane Sets 66 4.3 An Example: The Simple Pendulum (Part II) 68 4.4 Mather's Approach: Peierls' Barrier 71 5 The Hamilton-Jacobi Equation and Weak KAM Theory 76 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory 76 5.2 Regularity of Critical Subsolutions 85 5.3 Non-Wandering Points of the Mane Set 87 Appendices A On the Existence of Invariant Lagrangian Graphs 89 A.1 Symplectic Geometry of the Phase Space 89 A.2 Existence and Nonexistence of Invariant Lagrangian Graphs 91 B Schwartzman Asymptotic Cycle and Dynamics 97 B.1 Schwartzman Asymptotic Cycle 97 B.2 Dynamical Properties 99 Bibliography 107 Index 113

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book SynopsisA unique approach to teaching particle and rigid body dynamics using solved illustrative examples and exercises to encourage self-learning The study of particle and rigid body dynamics is a fundamental part of curricula for students pursuing graduate degrees in areas involving dynamics and control of systems. These include physics, robotics, nonlinear dynamics, aerospace, celestial mechanics and automotive engineering, among others. While the field of particle and rigid body dynamics has not evolved significantly over the past seven decades, neither have approaches to teaching this complex subject. This book fills the void in the academic literature by providing a uniquely stimulating, flipped classroom approach to teaching particle and rigid body dynamics which was developed, tested and refined by the author and his colleagues over the course of many years of instruction at both the graduate and undergraduate levels. Complete with numerous solved illustrTable of ContentsList of Figures xiii Preface xxiii Acknowledgement xxvii Introduction xxix About the Companion Website xliii 1 Kinematics of Particles 1 1.1 Inertial Frames 1 1.2 Rotating Frames 2 1.3 Rotation Matrices 4 1.4 Velocity of a Particle in a Three-dimensional Space 8 1.5 Acceleration of a Particle in a Three-dimensional Space 14 Exercises 21 2 Dynamics of Particles: Vectorial Approach 27 2.1 Newton’s Second Law of Dynamics 27 2.2 Stiffness and Viscous Damping 37 2.3 Dry Friction 40 2.4 Dynamics of a System of Particles 43 2.5 Newton’s Law of Gravitation 47 Exercises 50 Reference 54 3 Dynamics of Rigid Bodies: Vectorial Approach 55 3.1 Center of Mass 55 3.2 Mass Moment of Inertia 57 3.3 Parallel Axis Theorem 61 3.4 Rotation of the Inertia Matrix 65 3.4.1 The Principal Axes 66 3.5 Planar Motion of Rigid Bodies 69 3.5.1 Moment about an Inertial Point 72 3.5.2 Moment about a Moving Point on the Body 73 3.5.3 Moment about the Center of Mass or a Fixed Point on the Body 73 3.6 Non-planar Rigid-body Motion 83 3.6.1 Euler Rotational Equations 85 Exercises 94 Reference 101 4 System Constraints and Virtual Displacement 103 4.1 Constraints 103 4.1.1 Classification of Constraints 104 4.2 Actual and Virtual Displacements 110 4.3 Virtual Work 113 Exercises 115 Reference 116 5 Dynamics of Particles: Analytical Approach 117 5.1 The Brachistochrone Problem 117 5.2 Lagrange’s Equation for a Conservative System 123 5.3 Lagrange’s Equation for Non-conservative Systems 131 5.3.1 Viscous Damping 134 5.4 Lagrange’s Equations with Constraints 141 5.4.1 Physical Interpretation of Lagrange Multipliers 146 5.5 Cyclic Coordinates 151 5.6 Advantages and Disadvantages of the Analytical Approach 154 Exercises 155 References 159 6 Dynamics of Rigid Bodies: Analytical Approach 161 6.1 Kinetic Energy of a Rigid Body 161 6.2 Lagrange’s Equation Applied to Rigid Bodies 166 Exercises 176 7 Momentum 183 7.1 Linear Momentum 183 7.2 Collision 186 7.3 Angular Momentum of Particles 192 7.3.1 Angular Impulse 195 7.4 Angular Momentum of Rigid Bodies (Planar Motion) 199 7.4.1 Angular Momentum about an Axis Passing through the Center of Mass 199 7.4.2 Angular Momentum about an Axis Passing through a Fixed Point on the Body 201 7.4.3 Angular Momentum about an Axis Passing through an Arbitrary Inertial Point 201 7.5 Angular Momentum of Rigid Bodies (Non-planar Motion) 205 7.5.1 Angular Momentum about a Set of Axes Located at the Center of Mass 205 7.5.2 Angular Momentum about a Set of Axes Located at a Fixed Point 206 7.5.3 Angular Momentum about a Set of Axes Located at an Arbitrary Inertial Point 206 7.5.4 Conservation of Angular Momentum for Rigid Bodies 206 7.6 Generalized Momenta 213 Exercises 219 8 Motion of Charged Bodies in an Electric Field 227 8.1 Electrostatics 227 8.1.1 Electrostatic Forces 227 8.1.2 Electric Field 229 8.1.3 Electric Flux 232 8.1.4 Electrostatic Potential Energy 234 8.1.5 Electric Potential (Voltage) 235 8.1.6 Capacitance 237 8.1.7 Motion in an Electric Field 239 8.2 Electromagnetism 247 8.2.1 Electromagnetic Force 247 8.2.2 Forces on a Current-carrying Conductor 253 8.2.3 Electromagnetic Coupling 255 8.2.4 Ampere’s Law 257 8.2.5 Faraday’s Law of Induction 262 8.3 Lagrangian Formulation for Electrical Elements 268 8.3.1 Capacitor 268 8.3.2 Inductor 269 8.3.3 Resistor 269 8.4 Maxwell’s Equations 273 8.4.1 Maxwell’s First Equation 273 8.4.2 Maxwell’s Second Equation 273 8.4.3 Maxwell’s Third Equation 274 8.4.4 Maxwell’s Fourth Equation 274 8.5 Lagrangian Formulation of the Lorentz Force 275 Exercises 279 References 284 9 Introduction to Analysis Tools 285 9.1 Basic Definitions 285 9.2 Equilibrium Solutions of Dynamical Systems 287 9.3 Stability and Classification of Equilibrium Solutions 288 9.4 Phase-plane Representation of the Dynamics 296 9.4.1 Conservative Systems 296 9.4.2 Non-conservative Systems 303 9.5 Bifurcation of Equilibrium Solutions 308 9.5.1 Static Bifurcations 308 9.5.2 Dynamic (Hopf) Bifurcation 315 9.6 Basins of Attraction 323 Exercises 324 References 326 Index 327

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