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Titel:	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions
Von:	Florian Scheck
Person:	Scheck, Florian 1936-2024 Verfasser aut
Hauptverfasser:	Scheck, Florian 1936-2024 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Berlin [u.a.] Springer 2005
Ausgabe:	4. ed.
Notation:	UF 1000
Schlagwörter:	Mechanik Lehrbuch
Medienzugang:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012919667&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Dt. Ausg. u.d.T.: Scheck, Florian: Mechanik
Umfang:	XIV, 547 S. graph. Darst.
ISBN:	3540219250

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adam_text	Contents 1. Elementary Newtonian Mechanics ............................. 1 1.1 Newton s Laws (1687) and Their Interpretation ................ ] 1.2 Uniform Rectilinear Motion and Inertia! Systems ............... 4 1.3 Inerţial Frames in Relative Motion ........................... 6 1.4 Momentum and Force ...................................... 6 1.5 Typical Forces. A Remark About Units ....................... 8 1.6 Space, Time, and Forces .................................... 10 1.7 The Two-Body System with Internal Forces ................... Π 1.7.1 Center-of-Mass and Relative Motion ................... 11 1.7.2 Example: The Gravitational Force Between Two Celestial Bodies (Kepler s Problem) ....... 13 1.7.3 Center-of-Mass and Relative Momentum in the Two-Body System ............................. 19 1.8 Systems of Finitely Many Particles ........................... 20 1.9 The Principle of Center-of-Mass Motion ...................... 21 1.10 The Principle of Angular-Momentum Conservation ............. 21 1.11 The Principle of Energy Conservation ......................... 22 1.12 The Closed n-Particle System ................................ 23 1.13 Galilei Transformations ..................................... 24 1.14 Space and Time with Galilei Invariance ....................... 27 1.15 Conservative Force Fields ................................... 29 1.16 One-Dimensional Motion of a Point Particle ................... 32 1.17 Examples of Motion in One Dimension ....................... 33 1.17.1 The Harmonic Oscillator ............................. 33 1.17.2 The Planar Mathematical Pendulum .................... 35 1.18 Phase Space for the n-Particle System (in R3) .................. 37 1.19 Existence and Uniqueness of the Solutions of χ = F{x, t) ....... 38 1.20 Physical Consequences of the Existence and Uniqueness Theorem 39 1.21 Linear Systems ............................................ 41 1.21.1 Linear, Homogeneous Systems ........................ 41 1.21.2 Linear, Inhomogeneous Systems ....................... 42 1.22 Integrating One-Dimensional Equations of Motion .............. 42 1.23 Example: The Planar Pendulum for Arbitrary Deviations from the Vertical ........................................... 44 X Contents 1.24 Example: The Two-Body System with a Central Force .......... 47 1.25 Rotating Reference Systems: Coriolis and Centrifugal Forces .... 54 1.26 Examples of Rotating Reference Systems ..................... 55 1.27 Scattering of Two Particles that Interact via a Central Force: Kinematics ................................................ 63 1.28 Two-Particle Scattering with a Central Force: Dynamics ......... 67 1.29 Example: Coulomb Scattering of Two Particles with Equal Mass and Charge ................................ 71 1.30 Mechanical Bodies of Finite Extension ........................ 75 1.31 Time Averages and the Virial Theorem ....................... 79 Appendix: Practical Examples .................................... 81 2. The Principles of Canonical Mechanics ........................ 87 2.1 Constraints and Generalized Coordinates ...................... 87 2.1.1 Definition of Constraints ............................. 87 2.1.2 Generalized Coordinates .............................. 89 2.2 D Alembert s Principle ...................................... 89 2.2.1 Definition of Virtual Displacements .................... 89 2.2.2 The Static Case ..................................... 90 2.2.3 The Dynamical Case ................................. 90 2.3 Lagrange s Equations ....................................... 92 2.4 Examples of the Use of Lagrange s Equations .................. 93 2.5 A Digression on Variational Principles ........................ 95 2.6 Hamilton s Variational Principle (1834)....................... 98 2.7 The Euler-Lagrange Equations ............................... 98 2.8 Further Examples of the Use of Lagrange s Equations ........... 99 2.9 A Remark About Nonuniqueness of the Lagrangian Function ___101 2.10 Gauge Transformations of the Lagrangian Function .............102 2.11 Admissible Transformations of the Generalized Coordinates .....103 2.12 The Hamiltonian Function and Its Relation to the Lagrangian Function L ................................104 2.13 The Legendre Transformation for the Case of One Variable ......105 2.14 The Legendre Transformation for the Case of Several Variables .. 107 2.15 Canonical Systems .........................................108 2.16 Examples of Canonical Systems ..............................109 2.17 The Variational Principle Applied to the Hamiltonian Function ... Ill 2.18 Symmetries and Conservation Laws ..........................112 2.19 Noether s Theorem .........................................113 2.20 The Generator for Infinitesimal Rotations About an Axis ........115 2.21 More About the Rotation Group .............................117 2.22 Infinitesimal Rotations and Their Generators ...................119 2.23 Canonical Transformations ..................................121 2.24 Examples of Canonical Transformations .......................125 2.25 The Structure of the Canonical Equations .....................126 2.26 Example: Linear Autonomous Systems in One Dimension .......127 Contents XI 2.27 Canonical Transformations in Compact Notation ...............129 2.28 On the Symplectic Structure of Phase Space ...................131 2.29 Liouvilie s Theorem ........................................134 2.29.1 The Local Form .....................................135 2.29.2 The Global Form ....................................136 2.30 Examples for the Use of Liouvilie s Theorem ..................137 2.31 Poisson Brackets ...........................................140 2.32 Properties of Poisson Brackets ...............................143 2.33 Infinitesimal Canonical Transformations .......................145 2.34 Integrals of the Motion .....................................146 2.35 The Hamilton-Jacobi Differential Equation ....................149 2.36 Examples for the Use of the Hamilton-Jacobi Equation .........150 2.37 The Hamilton-Jacobi Equation and Integrable Systems ..........154 2.37.1 Local Rectification of Hamiltonian Systems .............154 2.37.2 Integrable Systems ...................................158 2.37.3 Angle and Action Variables ...........................163 2.38 Perturbing Quasiperiodic Hamiltonian Systems .................164 2.39 Autonomous, Nondegenerate Hamiltonian Systems in the Neighborhood of Integrable Systems .................... 1 67 2.40 Examples. The Averaging Principle ...........................168 2.40.1 The Anharmonic Oscillator ...........................168 2.40.2 Averaging of Perturbations ............................170 2.41 Generalized Theorem of Noether .............................172 Appendix: Practical Examples ....................................180 3. The Mechanics of Rigid Bodies ............................... 185 3.1 Definition of Rigid Body ....................................185 3.2 Infinitesimal Displacement of a Rigid Body ...................187 3.3 Kinetic Energy and the Inertia Tensor .........................189 3.4 Properties of the Inertia Tensor ..............................191 3.5 Steiner s Theorem ..........................................195 3.6 Examples of the Use of Steiner s Theorem ....................196 3.7 Angular Momentum of a Rigid Body .........................201 3.8 Force-Free Motion of Rigid Bodies ...........................203 3.9 Another Parametrization of Rotations: The Euler Angles ........205 3.10 Definition of Euierian Angles ................................207 3.11 Equations of Motion of Rigid Bodies .........................208 3.12 Euler s Equations of Motion .................................211 3.13 Euler s Equations Applied to a Force-Free Top .................214 3.14 The Motion of a Free Top and Geometric Constructions .........218 3.15 The Rigid Body in the Framework of Canonical Mechanics ......221 3.16 Example: The Symmetric Children s Top in a Gravitational Field .....................................225 3.17 More About the Spinning Top ...............................227 3.18 Spherical Top with Friction: The Tippe Top .................229 XII Contents 3.18.1 Conservation Law and Energy Considerations ...........230 3.18.2 Equations of Motion and Solutions with Constant Energy ................................232 Appendix: Practical Examples ....................................236 4. Relativistic Mechanics ....................................... 239 4.1 Failures of Nonrelativistic Mechanics .........................240 4.2 Constancy of the Speed of Light .............................243 4.3 The Lorentz Transformations ................................244 4.4 Analysis of Lorentz and Poincaré Transformations ..............250 4.4.1 Rotations and Special Lorentz Tranformations ( Boosts ) ..........................................252 4.4.2 Interpretation of Special Lorentz Transformations ........256 4.5 Decomposition of Lorentz Transformations into Their Components .....................................257 4.5.1 Proposition on Orthochronous, Proper Lorentz Transformations .......................257 4.5.2 Corollary of the Decomposition Theorem and Some Consequences ..............................259 4.6 Addition of Relativistic Velocities ............................262 4.7 Galilean and Lorentzian Space-Time Manifolds ................264 4.8 Orbital Curves and Proper Time .............................268 4.9 Relativistic Dynamics .......................................270 4.9.1 Newton s Equation ...................................270 4.9.2 The Energy-Momentum Vector ........................272 4.9.3 The Lorentz Force ...................................275 4.10 Time Dilatation and Scale Contraction ........................277 4.11 More About the Motion of Free Particles ......................279 4.12 The Conformai Group ......................................282 5. Geometric Aspects of Mechanics .............................. 283 5.1 Manifolds of Generalized Coordinates ........................284 5.2 Differentiable Manifolds ....................................287 5.2.1 The Euclidean Space W .............................287 5.2.2 Smooth or Differentiable Manifolds ....................289 5.2.3 Examples of Smooth Manifolds .......................291 5.3 Geometrical Objects on Manifolds ............................295 5.3.1 Functions and Curves on Manifolds ....................296 5.3.2 Tangent Vectors on a Smooth Manifold .................298 5.3.3 The Tangent Bundle of a Manifold .....................300 5.3.4 Vector Fields on Smooth Manifolds ....................301 5.3.5 Exterior Forms ......................................305 5.4 Calculus on Manifolds ......................................307 5.4.1 Differentiable Mappings of Manifolds ..................307 5.4.2 Integral Curves of Vector Fields .......................309 Contenis ХШ 5.4.3 Exterior Product of One-Forms ........................311 5.4.4 The Exterior Derivative ...............................313 5.4.5 Exterior Derivative and Vectors in R3 ..................315 5.5 Hamilton-Jacobi and Lagrangian Mechanics ...................317 5.5.1 Coordinate Manifold Q, Velocity Space TQ, and Phase Space T*Q ...............................317 5.5.2 The Canonical One-Form on Phase Space ...............321 5.5.3 The Canonical, Symplectic Two-Form on M ............324 5.5.4 Symplectic Two-Form and Darboux s Theorem ..........326 5.5.5 The Canonical Equations .............................329 5.5.6 The Poisson Bracket .................................332 5.5.7 Time-Dependent Hamiltonian Systems ..................335 5.6 Lagrangian Mechanics and Lagrange Equations ................337 5.6.1 The Relation Between the Two Formulations of Mechanics .......................................337 5.6.2 The Lagrangian Two-Form ............................339 5.6.3 Energy Function on Γ Q and Lagrangian Vector Field ___340 5.6.4 Vector Fields on Velocity Space T Q and Lagrange Equations ..............................342 5.6.5 The Legendre Transformation and the Correspondence of Lagrangian and Hamiltonian Functions ...............344 5.7 Riemannian Manifolds in Mechanics ..........................347 5.7.1 Affine Connection and Parallel Transport ...............348 5.7.2 Parallel Vector Fields and Geodesies ...................350 5.7.3 Geodesies as Solutions of Euler-Lagrange Equations .....351 5.7.4 Example: Force-Free Asymmetric Top ..................352 6. Stability and Chaos ......................................... 355 6.1 Qualitative Dynamics .......................................355 6.2 Vector Fields as Dynamical Systems ..........................356 6.2.1 Some Definitions of Vector Fields and Their Integral Curves .............................358 6.2.2 Equilibrium Positions and Linearization of Vector Fields .....................................360 6.2.3 Stability of Equilibrium Positions ......................363 6.2.4 Critical Points of Hamiltonian Vector Fields .............367 6.2.5 Stability and Instability of the Free Top ................369 6.3 Long-Term Behavior of Dynamical Flows and Dependence on External Parameters .....................................371 6.3.1 Flows in Phase Space ................................372 6.3.2 More General Criteria for Stability .....................373 6.3.3 Attractors ...........................................376 6.3.4 The Poincaré Mapping ...............................380 6.3.5 Bifurcations of Flows at Critical Points .................384 6.3.6 Bifurcations of Periodic Orbits ........................388 XIV Contents 6.4 Deterministic Chaos ........................................390 6.4.1 Iterative Mappings in One Dimension ..................390 6.4.2 Qualitative Definitions of Deterministic Chaos ...........392 6.4.3 An Example: The Logistic Equation ...................396 6.5 Quantitative Measures of Deterministic Chaos .................401 6.5.1 Routes to Chaos .....................................401 6.5.2 Liapunov Characteristic Exponents .....................405 6.5.3 Strange Attractors ...................................407 6.6 Chaotic Motions in Celestial Mechanics .......................409 6.6.1 Rotational Dynamics of Planetary Satellites .............409 6.6.2 Orbital Dynamics of Asteroids with Chaotic Behavior .... 415 7. Continuous Systems ......................................... 419 7.1 Discrete and Continuous Systems ............................419 7.2 Transition to the Continuous System ..........................423 7.3 Hamilton s Variational Principle for Continuous Systems ........425 7.4 Canonically Conjugate Momentum and Hamiltonian Density .....427 7.5 Example: The Pendulum Chain ..............................428 7.6 Comments and Outlook .....................................432 Exercises ...................................................... 437 Chapter 1 : Elementary Newtonian Mechanics .......................437 Chapter 2: The Principles of Canonical Mechanics ..................444 Chapter 3: The Mechanics of Rigid Bodies .........................452 Chapter 4: Relativistic Mechanics .................................455 Chapter 5: Geometric Aspects of Mechanics ........................458 Chapter 6: Stability and Chaos ...................................460 Solution of Exercises ............................................ 463 Chapter 1 : Elementary Newtonian Mechanics .......................463 Chapter 2: The Principles of Canonical Mechanics ..................478 Chapter 3: The Mechanics of Rigid Bodies .........................497 Chapter 4: Relativistic Mechanics .................................505 Chapter 5: Geometric Aspects of Mechanics ........................517 Chapter 6: Stability and Chaos ...................................522 Author Index ................................................... 541 Subject Index .................................................. 543
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spellingShingle	Scheck, Florian 1936-2024 Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions Mechanik (DE-588)4038168-7 gnd
subject_GND	(DE-588)4038168-7 (DE-588)4123623-3
title	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions
title_auth	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions
title_exact_search	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions
title_full	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions Florian Scheck
title_fullStr	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions Florian Scheck
title_full_unstemmed	Mechanics from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions Florian Scheck
title_short	Mechanics
title_sort	mechanics from newton s laws to deterministic chaos with 174 figures and 117 problems with complete solutions
title_sub	from Newton's laws to deterministic chaos ; with 174 figures and 117 problems with complete solutions
topic	Mechanik (DE-588)4038168-7 gnd
topic_facet	Mechanik Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012919667&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT scheckflorian mechanicsfromnewtonslawstodeterministicchaoswith174figuresand117problemswithcompletesolutions

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