Holdings: Classical mechanics

Classical mechanics: an introduction

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Bibliographic Details
Main Author:	Strauch, Dieter (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2009
Subjects:	Mechanics Theoretische Mechanik Lehrbuch
Online Access:	Inhaltsverzeichnis
Physical Description:	XXI, 405 S. Ill., graph. Darst.
ISBN:	9783540736158

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adam_text	Contents Preface ........................................................ v 1 The Newtonian Mechanics of Point-Mass Systems: General Properties ........................................ 1 1.1 Point Masses, Open and Closed Systems .................... 1 1.1.1 The Point Mass as an Idealization ................... 1 1.1.2 Open and Closed Systems, Internal and External Forces 2 1.2 Newton s Axioms (1687): A New Era ...................... 4 1.3 Einstein s Equivalence Principle (1916): Another New Era .... 8 1.3.1 Mass, Weight, Force, and the Like ................... 8 1.3.2 Einstein s Equivalence Principle ..................... 9 1.4 Types of Forces ......................................... 9 1.4.1 Central Forces .................................... 9 1.4.2 Conservative Forces (I) ............................. 11 1.4.3 Central Potentials ................................. 12 1.4.4 Potential and Potential Energy ...................... 12 1.4.5 The Lorentz Force ................................. 12 1.4.6 Frictional Forces .................................. 13 1.5 Equations of Motion ..................................... 14 1.5.1 Definitions ....................................... 14 1.5.2 Equations of Motion ............................... 15 1.6 Conservation Laws ...................................... 17 1.6.1 Conservation of Momentum ......................... 17 1.6.2 Angular-Momentum Conservation and Central Forces .. 18 1.7 Energy-Conservation Theorem and Conservative Forces ...... 18 1.7.1 Conservative Forces (II) ............................ 18 1.7.2 Potential Energy, Work, and Power .................. 19 1.7.3 Energy Conservation Theorem ...................... 20 1.8 Invariances and Conservation Laws ........................ 20 1.8.1 Translations and Rotations ......................... 20 Contents 1.8.2 Invariances and Conservation Laws: Noether s Theorem (I) ............................. 21 1.8.3 The Usefulness of the Conservation Laws ............. 23 1.8.4 The Usefulness of Symmetry Analysis ................ 23 1.9 * Virial Theorem ........................................ 23 1.10 The Basic Problem of the Mechanics of a Point Mass ........ 25 Summary: Newtonian Mechanics - General Properties ............ 25 Problems ................................................... 27 Newtonian Mechanics: First Applications .................. 29 2.1 One-Dimensional Motion ................................. 29 2.1.1 Constant Force .................................... 29 2.1.2 Time-Dependent Force ............................. 30 2.1.3 Velocity-Dependent Force .......................... 31 2.1.4 Coordinate-Dependent Force ........................ 32 2.1.5 Example: Plane Pendulum (with Small Amplitude) .... 34 2.1.6 * Example: Plane Pendulum with Large Amplitude .... 38 2.2 Motion in a Plane ....................................... 41 2.2.1 Fixed and Moving Basis ............................ 41 2.2.2 Time Dependence of the Moving Basis ............... 42 2.2.3 Velocity, Acceleration, etc. in Plane Polar Coordinates . 44 2.2.4 Example: Plane Pendulum .......................... 46 2.3 Two-Particle Systems .................................... 48 2.3.1 Center-of-Mass and Relative Coordinates ............. 48 2.3.2 Equations of Motion ............................... 49 2.3.3 Closed Systems ................................... 50 Summary: Newtonian Mechanics - First Applications ............. 51 Problems ................................................... 52 Lagrangian Mechanics ..................................... 55 3.1 Motion with Constraints ................................. 56 3.2 Constraints and Generalized Coordinates ................... 57 3.2.1 Examples of Constraints ........................... 57 3.2.2 Classification of Constraints ........................ 58 3.2.3 Degrees of Freedom ................................ 61 3.2.4 Generalized Coordinates ........................... 62 3.2.5 Example: Plane Double Pendulum ................... 62 3.3 Forces of Constraint ..................................... 63 3.4 The Principle of d Alembert .............................. 64 3.4.1 Real and Virtual Displacements ..................... 64 3.4.2 The Principle of Virtual Work ...................... 66 3.4.3 The Principle of d Alembert ........................ 67 3.5 Lagrangian Equations of the Second Kind .................. 67 3.5.1 External and Internal Forces ........................ 68 3.5.2 Conservative Forces ................................ 68 Contents xi 3.5.3 Inerţial Forces.................................... 69 3.5.4 Lagrangian Equations of Motion .................... 70 3.5.5 Example: Plane Motion in a Central Potential........ 72 3.5.6 Example: Bead Sliding on a Uniformly Rotating Rod ..................................... 73 3.5.7 Example: Plane Double Pendulum ................... 74 3.5.8 Separable Systems ................................. 76 3.6 Generalized Momentum, Force, etc ......................... 78 3.6.1 Generalized Force and Generalized Momentum ........ 78 3.7 Velocity-Dependent Forces ................................ 80 3.7.1 Generalized Potential .............................. 80 3.7.2 Lorentz Force ..................................... 80 3.8 Gauge Invariance and Form Invariance ..................... 81 3.8.1 (Un)Ambiguity of the Lagrangian Function: Gauge Transformation ............................. 81 3.8.2 * Form Invariance of the Lagrangian Equations under Point Transformations ........................ 83 3.9 * Lagrangian Equations of the First Kind .................. 84 3.9.1 Lagrange Multipliers ............................... 84 3.9.2 Lagrangian Equations of the First Kind .............. 85 3.9.3 Example: Atwood Machine ......................... 86 3.10 Hamilton s Principle ..................................... 92 3.10.1 The Action ....................................... 92 3.10.2 Hamilton s Principle ............................... 93 3.11 Symmetries and Conservation Laws ........................ 95 3.11.1 Noether s Theorem (II) ............................ 95 3.11.2 Cyclic Coordinate and Conservation of the Conjugate Momentum ........................ 96 3.11.3 Homogeneity in Time and Energy Conservation ....... 97 3.11.4 Space Homogeneity and Conservation of Momentum ... 100 3.11.5 Isotropy and Angular-Momentum Conservation .......100 Summary: Lagrangian Mechanics ..............................101 Problems ...................................................102 4 Harmonic Vibrations ......................................113 4.1 The Simple Oscillator ....................................117 4.1.1 Eigen Solutions of the Homogeneous Differential Equation .........................................117 4.2 The Free Oscillator with Damping .........................120 4.2.1 Eigen Solutions of the Homogeneous Differential Equation .........................................120 4.2.2 Reality of the Solution .............................122 4.2.3 Initial Conditions .................................123 Contents 4.3 The Forced Harmonic Oscillator ...........................125 4.3.1 Equation of Motion ................................125 4.3.2 Solution of the Inhomogeneous Linear Differential Equation by Fourier Transformation .................126 4.3.3 Green Function of the Damped Oscillator in Frequency Space ................................129 4.3.4 Time-Dependent Green Function ....................131 4.3.5 Energy Considerations .............................134 4.4 Coupled Oscillators ......................................138 4.4.1 Introductory Example: Stretching Vibrations in the CO2 Molecule ...............................138 4.4.2 General Coupled Vibrations ........................145 4.4.3 Normal Coordinates ...............................147 Summary: Harmonic Vibrations ................................148 Problems ...................................................149 Central Potentials and the Kepler Problem ................157 5.1 Central Force and Motion in a Plane .......................157 5.1.1 Central Potential, Central Force, and Angular-Momentum Conservation ...................157 5.1.2 Central Potential and Effective Radial Potential .......159 5.1.3 Central Potential and Trajectory ....................160 5.2 Kinematics of the Kepler Motion ..........................163 5.2.1 Kepler s Laws .....................................163 5.2.2 Polar Representation of the Conies ..................165 5.2.3 Determination of the Force and Potential from the Trajectory ................................166 5.2.4 Determination of the Trajectory from the Potential .... 168 5.2.5 Trajectory and Rotation Periods ....................171 5.2.6 The Laplace-Runge-Lenz Vector ....................173 5.2.7 Perihelion Rotation ................................175 Summary: Central Potentials and the Kepler Problem ............177 Problems ...................................................178 Collision and Scattering Problems .........................183 6.1 Kinematics .............................................183 6.1.1 Scattering and Collision ............................184 6.1.2 Momentum Change and Impulsive Force .............184 6.1.3 Laboratory System and Center-of-Mass System .......185 6.1.4 Consequences of the Conservation of Momentum ......186 6.1.5 Elastic and Inelastic Scattering ......................187 6.1.6 Consequences of the Angular-Momentum Conservation .....................................189 Contents xiii 6.2 Collision of Hard Spheres .................................190 6.2.1 Notations ........................................190 6.2.2 Elastic Collision of Smooth Spheres (Laboratory System) ...............................190 6.3 Scattering by a Central Potential ..........................194 6.3.1 Central Potential and Scattering Angle ...............194 6.3.2 Scattering by the Gravitational Potential .............195 6.4 * The Cross-Section .....................................196 6.4.1 The Problem .....................................196 6.4.2 Scattering of Many Probe Particles by Many Target Particles ...........................197 6.4.3 Scattering from a Particle at Rest ...................197 6.4.4 The Differential Cross-Section .......................198 6.4.5 The Total Cross-Section ............................200 6.4.6 The Rutherford Cross-Section .......................200 6.4.7 Scattering Cross-Section for a Collision of Hard Spheres ...................................201 Summary: Collision and Scattering Problems ....................202 Problems ...................................................203 Moving Reference Frames .................................207 7.1 Translations ............................................208 7.1.1 The Transformation ...............................208 7.1.2 Inerţial Forces ....................................209 7.2 Rotation Around a Fixed Point ...........................209 7.2.1 Active and Passive Rotation ........................209 7.2.2 Infinitesimal Rotations .............................210 7.2.3 Representation in Different Coordinate Systems .......211 7.2.4 Uniformly Rotating System: Centrifugal and Coriolis Force .................................214 7.2.5 The Foucault Pendulum ............................214 7.3 Galilean and Lorentz Transformation ......................217 7.3.1 The Relativity Principle ............................218 7.3.2 General and Special Transformation .................218 7.3.3 The Galilean Transformation .......................219 7.3.4 Galilean Invariance ................................220 7.3.5 The Lorentz Transformation ........................221 Summary: Moving Reference Frames ...........................222 Problems ...................................................223 Dynamics of a Rigid Body .................................227 8.1 The Rigid Body as a System of N Point Masses .............227 8.2 Translational and Rotational Energy: Inertia Tensor .........228 8.2.1 The Center of Mass as the Specific Point .............228 8.2.2 The Instantaneously Fixed Point as the Specific Point .. 231 Contents 8.3 Transition to the Continuum ..............................232 8.3.1 Example: Center of Mass of a Homogeneous Hemisphere .......................................233 8.3.2 Examples of Inertia Tensors ........................234 8.4 Change of the Reference System: Steiner s Theorem ..........238 8.4.1 Steiner s Theorem .................................238 8.4.2 Angular Momentum and Torque ....................239 8.4.3 Examples ........................................241 8.5 Principal Moments of Inertia and Principal Axes ............242 8.5.1 Definitions .......................................242 8.5.2 Examples ........................................244 8.6 Rotation Around a Fixed Axis ............................247 8.6.1 Inertia Tensor and Moment of Inertia ................248 8.6.2 The Equation of Motion ............................249 8.6.3 Example: Motion in the Homogeneous Gravitational Field: The Physical Pendulum ......................249 8.6.4 Example: A Cylinder Rolling on an Inclined Plane .....251 8.7 Rotation Around a Fixed Point: The Top ...................253 8.7.1 Space-Fixed (Inerţial) and Body-Fixed Coordinate System ...........................................253 8.7.2 * Euler Angles ....................................254 8.7.3 * The Euler Equations of Motion ....................257 8.8 The Force-Free Symmetrical Top ..........................258 8.8.1 The Equations of Motion ...........................258 8.8.2 * Stability of the Rotational Motion of the Top .......260 Summary: Dynamics of a Rigid Body ...........................262 Problems ...................................................263 Hamiltonian Dynamics ....................................269 9.1 Hamiltonian Equations of Motion ..........................269 9.1.1 The Hamiltonian Function ..........................269 9.1.2 Canonical (Hamiltonian) Equations of Motion .........271 9.1.3 * (Un)Ambiguity of the Hamiltonian Function: Gauge Transformation .............................274 9.2 * Poisson Brackets .......................................275 9.2.1 Definition 29. (Poisson Brackets) ....................275 9.2.2 Equations of Motion ...............................276 9.2.3 Fundamental Poisson Brackets ......................277 9.2.4 Properties of the Poisson Brackets ...................277 9.2.5 Example: Harmonic oscillator .......................277 9.2.6 Transition to Quantum Mechanics ...................278 9.3 Configuration Space and Phase Space ......................278 9.3.1 Configuration Space ...............................278 9.3.2 Phase Space ......................................279 Contents xv 9.4 * The Modified Hamilton Principle ........................281 9.5 * Canonical Transformation ...............................282 9.5.1 Point Transformation in Configuration Space .........282 9.5.2 Point Transformation in Phase Space ................283 9.5.3 Canonical Transformation ..........................284 9.5.4 Time Development as a Canonical Transformation .....287 9.5.5 Canonical Invariants ...............................288 9.5.6 Generating Function ...............................289 9.5.7 The Theorem of Liouville ...........................294 9.6 * Hamilton-Jacobi Equation ..............................296 9.6.1 Action Function ...................................296 9.6.2 The Characteristic Function ........................298 9.6.3 Examples ........................................299 Summary: Hamiltonian Dynamics ..............................303 Problems ...................................................304 10 * Introduction to the Mechanics of Continua ..............309 10.1 Fields ..................................................309 10.2 Lagrangian Density ......................................311 10.2.1 Kinetic Energy Density ............................311 10.2.2 Potential Energy Density ...........................312 10.2.3 Lagrangian Density ................................314 10.3 Hamilton s Principle and Lagrangian Equations .............314 10.4 The Energy-Momentum Tensor ...........................317 10.4.1 The Conservation Theorem .........................317 10.4.2 Energy Density ...................................318 10.4.3 Energy-Current Density ............................319 10.4.4 Energy Conservation ...............................320 10.4.5 The Strain-Momentum Density .....................320 10.4.6 The Stress Tensor .................................322 10.4.7 Conservation of the Strain Momentum ...............323 Summary: Mechanics of Continua ..............................324 A Physical Constants ........................................325 В Scalars, Vectors, Tensors ..................................329 B.I Definitions and Simple Rules ..............................329 B.I.I Definitions .......................................329 B.1.2 Behavior Under Inversion ...........................330 B.2 Vectors .................................................330 B.2.1 Rules for Vectors ..................................330 B.2.2 Dyadic Product (Tensor Product) of Vectors ..........331 B.2.3 Vector Product (Cross Product) of Vectors ...........331 B.2.4 Triple Scalar Product of Vectors .....................332 B.2.5 Multiple Products of Vectors ........................332 xvi Contents С Rectangular Coordinate Systems ..........................333 Cl Definitions ..............................................333 С. 2 Cartesian Coordinates ...................................333 C.3 Spherical Polar Coordinates (Spherical Coordinates) .........334 C.4 Cylindrical Coordinates ..................................335 C.5 Plane Polar Coordinates ..................................335 C.6 Inverse Relations ........................................336 Problems ...................................................336 D Nabla (Del) Operator and Laplace Operator ...............339 D.I Representations of the Nabla and Laplace Operators .........339 D.2 Standard Relations ......................................340 D.3 Rules ..................................................341 D.4 Integral Theorems .......................................342 D.4.1 The Theorem of Gauß .............................342 D.4.2 Stokes Theorem ..................................343 D.4.3 The Theorem of Green .............................343 Problems ...................................................344 E Variational Method ........................................347 E.I Functions and Functionals ................................347 E.2 Variational Problem and Euler Equation ...................348 Problems ...................................................350 F Linear Differential Equations with Constant Coefficients ... 351 F.I Homogeneous Linear Differential Equations .................351 F.2 Inhomogeneous Linear Differential Equations ...............352 F.3 Stability of Solutions .....................................353 G Quadratic Matrices and Their Eigen Solutions .............355 G.I The Eigen Value Problem ................................355 G.2 Definitions ..............................................355 G.3 Properties of the Eigen Values ............................357 G.4 Properties of the Eigen Vectors of Hermitian Matrices ........358 H Dirac б- Function and Heaviside Step Function .............361 H.I Properties of the Dirac ¿-Function and of the Heaviside Step Function ...........................................361 H.2 Representation of the ¿-Function by Functional Sequences .... 362 H.3 Integral Representation of the ¿»-Function ...................363 H.4 Periodic ¿-Function ......................................363 H.5 The ¿-Function in R3 ....................................363 H.6 The ¿-Function as an Inhomogeneity of the Poisson Equation . 364 Contents xvii I Fourier Transformation ....................................365 1.1 The Transformation: Fourier Integral ......................365 1.1.1 Examples and Applications .........................366 1.1.2 Convolution Theorem ..............................370 1.1.3 Parsevaľs Equation ................................371 1.1.4 Uncertainty Relation ...............................372 1.2 Fourier Transformation in IR4: Plane Waves .................374 1.2.1 The Whole K3 ....................................374 1.2.2 Normalization Volume V ...........................375 1.3 Fourier Series ...........................................376 1.3.1 The Series ........................................376 1.3.2 Examples and Applications .........................378 1.3.3 Convolution Theorem ..............................380 1.3.4 Parsevaľs Equation ................................382 1.3.5 Fourier Series in Ж3: Lattices .......................382 1.3.6 Functions with Lattice Periodicity ...................383 Summary: Fourier Integral and Fourier Series ....................384 Problems ...................................................385 J Change of Variables: Legendre Transformation .............387 References .....................................................389 Index ..........................................................391
adam_txt	Contents Preface . v 1 The Newtonian Mechanics of Point-Mass Systems: General Properties . 1 1.1 Point Masses, Open and Closed Systems . 1 1.1.1 The Point Mass as an Idealization . 1 1.1.2 Open and Closed Systems, Internal and External Forces 2 1.2 Newton's Axioms (1687): A New Era . 4 1.3 Einstein's Equivalence Principle (1916): Another New Era . 8 1.3.1 Mass, Weight, Force, and the Like . 8 1.3.2 Einstein's Equivalence Principle . 9 1.4 Types of Forces . 9 1.4.1 Central Forces . 9 1.4.2 Conservative Forces (I) . 11 1.4.3 Central Potentials . 12 1.4.4 Potential and Potential Energy . 12 1.4.5 The Lorentz Force . 12 1.4.6 Frictional Forces . 13 1.5 Equations of Motion . 14 1.5.1 Definitions . 14 1.5.2 Equations of Motion . 15 1.6 Conservation Laws . 17 1.6.1 Conservation of Momentum . 17 1.6.2 Angular-Momentum Conservation and Central Forces . 18 1.7 Energy-Conservation Theorem and Conservative Forces . 18 1.7.1 Conservative Forces (II) . 18 1.7.2 Potential Energy, Work, and Power . 19 1.7.3 Energy Conservation Theorem . 20 1.8 Invariances and Conservation Laws . 20 1.8.1 Translations and Rotations . 20 Contents 1.8.2 Invariances and Conservation Laws: Noether's Theorem (I) . 21 1.8.3 The Usefulness of the Conservation Laws . 23 1.8.4 The Usefulness of Symmetry Analysis . 23 1.9 * Virial Theorem . 23 1.10 The Basic Problem of the Mechanics of a Point Mass . 25 Summary: Newtonian Mechanics - General Properties . 25 Problems . 27 Newtonian Mechanics: First Applications . 29 2.1 One-Dimensional Motion . 29 2.1.1 Constant Force . 29 2.1.2 Time-Dependent Force . 30 2.1.3 Velocity-Dependent Force . 31 2.1.4 Coordinate-Dependent Force . 32 2.1.5 Example: Plane Pendulum (with Small Amplitude) . 34 2.1.6 * Example: Plane Pendulum with Large Amplitude . 38 2.2 Motion in a Plane . 41 2.2.1 Fixed and Moving Basis . 41 2.2.2 Time Dependence of the Moving Basis . 42 2.2.3 Velocity, Acceleration, etc. in Plane Polar Coordinates . 44 2.2.4 Example: Plane Pendulum . 46 2.3 Two-Particle Systems . 48 2.3.1 Center-of-Mass and Relative Coordinates . 48 2.3.2 Equations of Motion . 49 2.3.3 Closed Systems . 50 Summary: Newtonian Mechanics - First Applications . 51 Problems . 52 Lagrangian Mechanics . 55 3.1 Motion with Constraints . 56 3.2 Constraints and Generalized Coordinates . 57 3.2.1 Examples of Constraints . 57 3.2.2 Classification of Constraints . 58 3.2.3 Degrees of Freedom . 61 3.2.4 Generalized Coordinates . 62 3.2.5 Example: Plane Double Pendulum . 62 3.3 Forces of Constraint . 63 3.4 The Principle of d'Alembert . 64 3.4.1 Real and Virtual Displacements . 64 3.4.2 The Principle of Virtual Work . 66 3.4.3 The Principle of d'Alembert . 67 3.5 Lagrangian Equations of the Second Kind . 67 3.5.1 External and Internal Forces . 68 3.5.2 Conservative Forces . 68 Contents xi 3.5.3 Inerţial Forces. 69 3.5.4 Lagrangian Equations of Motion . 70 3.5.5 Example: Plane Motion in a Central Potential. 72 3.5.6 Example: Bead Sliding on a Uniformly Rotating Rod . 73 3.5.7 Example: Plane Double Pendulum . 74 3.5.8 Separable Systems . 76 3.6 Generalized Momentum, Force, etc . 78 3.6.1 Generalized Force and Generalized Momentum . 78 3.7 Velocity-Dependent Forces . 80 3.7.1 Generalized Potential . 80 3.7.2 Lorentz Force . 80 3.8 Gauge Invariance and Form Invariance . 81 3.8.1 (Un)Ambiguity of the Lagrangian Function: Gauge Transformation . 81 3.8.2 * Form Invariance of the Lagrangian Equations under Point Transformations . 83 3.9 * Lagrangian Equations of the First Kind . 84 3.9.1 Lagrange Multipliers . 84 3.9.2 Lagrangian Equations of the First Kind . 85 3.9.3 Example: Atwood Machine . 86 3.10 Hamilton's Principle . 92 3.10.1 The Action . 92 3.10.2 Hamilton's Principle . 93 3.11 Symmetries and Conservation Laws . 95 3.11.1 Noether's Theorem (II) . 95 3.11.2 Cyclic Coordinate and Conservation of the Conjugate Momentum . 96 3.11.3 Homogeneity in Time and Energy Conservation . 97 3.11.4 Space Homogeneity and Conservation of Momentum . 100 3.11.5 Isotropy and Angular-Momentum Conservation .100 Summary: Lagrangian Mechanics .101 Problems .102 4 Harmonic Vibrations .113 4.1 The Simple Oscillator .117 4.1.1 Eigen Solutions of the Homogeneous Differential Equation .117 4.2 The Free Oscillator with Damping .120 4.2.1 Eigen Solutions of the Homogeneous Differential Equation .120 4.2.2 Reality of the Solution .122 4.2.3 Initial Conditions .123 Contents 4.3 The Forced Harmonic Oscillator .125 4.3.1 Equation of Motion .125 4.3.2 Solution of the Inhomogeneous Linear Differential Equation by Fourier Transformation .126 4.3.3 Green Function of the Damped Oscillator in Frequency Space .129 4.3.4 Time-Dependent Green Function .131 4.3.5 Energy Considerations .134 4.4 Coupled Oscillators .138 4.4.1 Introductory Example: Stretching Vibrations in the CO2 Molecule .138 4.4.2 General Coupled Vibrations .145 4.4.3 Normal Coordinates .147 Summary: Harmonic Vibrations .148 Problems .149 Central Potentials and the Kepler Problem .157 5.1 Central Force and Motion in a Plane .157 5.1.1 Central Potential, Central Force, and Angular-Momentum Conservation .157 5.1.2 Central Potential and Effective Radial Potential .159 5.1.3 Central Potential and Trajectory .160 5.2 Kinematics of the Kepler Motion .163 5.2.1 Kepler's Laws .163 5.2.2 Polar Representation of the Conies .165 5.2.3 Determination of the Force and Potential from the Trajectory .166 5.2.4 Determination of the Trajectory from the Potential . 168 5.2.5 Trajectory and Rotation Periods .171 5.2.6 The Laplace-Runge-Lenz Vector .173 5.2.7 Perihelion Rotation .175 Summary: Central Potentials and the Kepler Problem .177 Problems .178 Collision and Scattering Problems .183 6.1 Kinematics .183 6.1.1 Scattering and Collision .184 6.1.2 Momentum Change and Impulsive Force .184 6.1.3 Laboratory System and Center-of-Mass System .185 6.1.4 Consequences of the Conservation of Momentum .186 6.1.5 Elastic and Inelastic Scattering .187 6.1.6 Consequences of the Angular-Momentum Conservation .189 Contents xiii 6.2 Collision of Hard Spheres .190 6.2.1 Notations .190 6.2.2 Elastic Collision of Smooth Spheres (Laboratory System) .190 6.3 Scattering by a Central Potential .194 6.3.1 Central Potential and Scattering Angle .194 6.3.2 Scattering by the Gravitational Potential .195 6.4 * The Cross-Section .196 6.4.1 The Problem .196 6.4.2 Scattering of Many Probe Particles by Many Target Particles .197 6.4.3 Scattering from a Particle at Rest .197 6.4.4 The Differential Cross-Section .198 6.4.5 The Total Cross-Section .200 6.4.6 The Rutherford Cross-Section .200 6.4.7 Scattering Cross-Section for a Collision of Hard Spheres .201 Summary: Collision and Scattering Problems .202 Problems .203 Moving Reference Frames .207 7.1 Translations .208 7.1.1 The Transformation .208 7.1.2 Inerţial Forces .209 7.2 Rotation Around a Fixed Point .209 7.2.1 Active and Passive Rotation .209 7.2.2 Infinitesimal Rotations .210 7.2.3 Representation in Different Coordinate Systems .211 7.2.4 Uniformly Rotating System: Centrifugal and Coriolis Force .214 7.2.5 The Foucault Pendulum .214 7.3 Galilean and Lorentz Transformation .217 7.3.1 The Relativity Principle .218 7.3.2 General and Special Transformation .218 7.3.3 The Galilean Transformation .219 7.3.4 Galilean Invariance .220 7.3.5 The Lorentz Transformation .221 Summary: Moving Reference Frames .222 Problems .223 Dynamics of a Rigid Body .227 8.1 The Rigid Body as a System of N Point Masses .227 8.2 Translational and Rotational Energy: Inertia Tensor .228 8.2.1 The Center of Mass as the Specific Point .228 8.2.2 The Instantaneously Fixed Point as the Specific Point . 231 Contents 8.3 Transition to the Continuum .232 8.3.1 Example: Center of Mass of a Homogeneous Hemisphere .233 8.3.2 Examples of Inertia Tensors .234 8.4 Change of the Reference System: Steiner's Theorem .238 8.4.1 Steiner's Theorem .238 8.4.2 Angular Momentum and Torque .239 8.4.3 Examples .241 8.5 Principal Moments of Inertia and Principal Axes .242 8.5.1 Definitions .242 8.5.2 Examples .244 8.6 Rotation Around a Fixed Axis .247 8.6.1 Inertia Tensor and Moment of Inertia .248 8.6.2 The Equation of Motion .249 8.6.3 Example: Motion in the Homogeneous Gravitational Field: The Physical Pendulum .249 8.6.4 Example: A Cylinder Rolling on an Inclined Plane .251 8.7 Rotation Around a Fixed Point: The Top .253 8.7.1 Space-Fixed (Inerţial) and Body-Fixed Coordinate System .253 8.7.2 * Euler Angles .254 8.7.3 * The Euler Equations of Motion .257 8.8 The Force-Free Symmetrical Top .258 8.8.1 The Equations of Motion .258 8.8.2 * Stability of the Rotational Motion of the Top .260 Summary: Dynamics of a Rigid Body .262 Problems .263 Hamiltonian Dynamics .269 9.1 Hamiltonian Equations of Motion .269 9.1.1 The Hamiltonian Function .269 9.1.2 Canonical (Hamiltonian) Equations of Motion .271 9.1.3 * (Un)Ambiguity of the Hamiltonian Function: Gauge Transformation .274 9.2 * Poisson Brackets .275 9.2.1 Definition 29. (Poisson Brackets) .275 9.2.2 Equations of Motion .276 9.2.3 Fundamental Poisson Brackets .277 9.2.4 Properties of the Poisson Brackets .277 9.2.5 Example: Harmonic oscillator .277 9.2.6 Transition to Quantum Mechanics .278 9.3 Configuration Space and Phase Space .278 9.3.1 Configuration Space .278 9.3.2 Phase Space .279 Contents xv 9.4 * The Modified Hamilton Principle .281 9.5 * Canonical Transformation .282 9.5.1 Point Transformation in Configuration Space .282 9.5.2 Point Transformation in Phase Space .283 9.5.3 Canonical Transformation .284 9.5.4 Time Development as a Canonical Transformation .287 9.5.5 Canonical Invariants .288 9.5.6 Generating Function .289 9.5.7 The Theorem of Liouville .294 9.6 * Hamilton-Jacobi Equation .296 9.6.1 Action Function .296 9.6.2 The Characteristic Function .298 9.6.3 Examples .299 Summary: Hamiltonian Dynamics .303 Problems .304 10 * Introduction to the Mechanics of Continua .309 10.1 Fields .309 10.2 Lagrangian Density .311 10.2.1 Kinetic Energy Density .311 10.2.2 Potential Energy Density .312 10.2.3 Lagrangian Density .314 10.3 Hamilton's Principle and Lagrangian Equations .314 10.4 The Energy-Momentum Tensor .317 10.4.1 The Conservation Theorem .317 10.4.2 Energy Density .318 10.4.3 Energy-Current Density .319 10.4.4 Energy Conservation .320 10.4.5 The Strain-Momentum Density .320 10.4.6 The Stress Tensor .322 10.4.7 Conservation of the Strain Momentum .323 Summary: Mechanics of Continua .324 A Physical Constants .325 В Scalars, Vectors, Tensors .329 B.I Definitions and Simple Rules .329 B.I.I Definitions .329 B.1.2 Behavior Under Inversion .330 B.2 Vectors .330 B.2.1 Rules for Vectors .330 B.2.2 Dyadic Product (Tensor Product) of Vectors .331 B.2.3 Vector Product (Cross Product) of Vectors .331 B.2.4 Triple Scalar Product of Vectors .332 B.2.5 Multiple Products of Vectors .332 xvi Contents С Rectangular Coordinate Systems .333 Cl Definitions .333 С. 2 Cartesian Coordinates .333 C.3 Spherical Polar Coordinates (Spherical Coordinates) .334 C.4 Cylindrical Coordinates .335 C.5 Plane Polar Coordinates .335 C.6 Inverse Relations .336 Problems .336 D Nabla (Del) Operator and Laplace Operator .339 D.I Representations of the Nabla and Laplace Operators .339 D.2 Standard Relations .340 D.3 Rules .341 D.4 Integral Theorems .342 D.4.1 The Theorem of Gauß .342 D.4.2 Stokes' Theorem .343 D.4.3 The Theorem of Green .343 Problems .344 E Variational Method .347 E.I Functions and Functionals .347 E.2 Variational Problem and Euler Equation .348 Problems .350 F Linear Differential Equations with Constant Coefficients . 351 F.I Homogeneous Linear Differential Equations .351 F.2 Inhomogeneous Linear Differential Equations .352 F.3 Stability of Solutions .353 G Quadratic Matrices and Their Eigen Solutions .355 G.I The Eigen Value Problem .355 G.2 Definitions .355 G.3 Properties of the Eigen Values .357 G.4 Properties of the Eigen Vectors of Hermitian Matrices .358 H Dirac б- Function and Heaviside Step Function .361 H.I Properties of the Dirac ¿-Function and of the Heaviside Step Function .361 H.2 Representation of the ¿-Function by Functional Sequences . 362 H.3 Integral Representation of the ¿»-Function .363 H.4 Periodic ¿-Function .363 H.5 The ¿-Function in R3 .363 H.6 The ¿-Function as an Inhomogeneity of the Poisson Equation . 364 Contents xvii I Fourier Transformation .365 1.1 The Transformation: Fourier Integral .365 1.1.1 Examples and Applications .366 1.1.2 Convolution Theorem .370 1.1.3 Parsevaľs Equation .371 1.1.4 Uncertainty Relation .372 1.2 Fourier Transformation in IR4: Plane Waves .374 1.2.1 The Whole K3 .374 1.2.2 Normalization Volume V .375 1.3 Fourier Series .376 1.3.1 The Series .376 1.3.2 Examples and Applications .378 1.3.3 Convolution Theorem .380 1.3.4 Parsevaľs Equation .382 1.3.5 Fourier Series in Ж3: Lattices .382 1.3.6 Functions with Lattice Periodicity .383 Summary: Fourier Integral and Fourier Series .384 Problems .385 J Change of Variables: Legendre Transformation .387 References .389 Index .391
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spelling	Strauch, Dieter Verfasser aut Classical mechanics an introduction Dieter Strauch Berlin [u.a.] Springer 2009 XXI, 405 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mechanics Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Theoretische Mechanik (DE-588)4185100-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-73616-5 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016234250&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Strauch, Dieter Classical mechanics an introduction Mechanics Theoretische Mechanik (DE-588)4185100-6 gnd
subject_GND	(DE-588)4185100-6 (DE-588)4123623-3
title	Classical mechanics an introduction
title_auth	Classical mechanics an introduction
title_exact_search	Classical mechanics an introduction
title_exact_search_txtP	Classical mechanics an introduction
title_full	Classical mechanics an introduction Dieter Strauch
title_fullStr	Classical mechanics an introduction Dieter Strauch
title_full_unstemmed	Classical mechanics an introduction Dieter Strauch
title_short	Classical mechanics
title_sort	classical mechanics an introduction
title_sub	an introduction
topic	Mechanics Theoretische Mechanik (DE-588)4185100-6 gnd
topic_facet	Mechanics Theoretische Mechanik Lehrbuch
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