A modern approach to classical mechanics:
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2016]
|
| Edition: | Second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xii, 515 Seiten Illustrationen, Diagramme |
| ISBN: | 9789814696289 9789814704113 |
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| adam_text | A MODERN APPROACH TO CLASSICAL MECHANICS
/ IRO, HARALD [AUTHOR.]
: 2015
TABLE OF CONTENTS / INHALTSVERZEICHNIS
THE FOUNDATIONS OF MECHANICS
ONE-DIMENSIONAL MOTION OF A PARTICLE
ENCOUNTERING PECULIAR MOTION IN TWO DIMENSIONS
MOTION IN A CENTRAL FORCE FIELD
THE GRAVITATIONAL INTERACTION OF TWO BODIES
COLLISIONS OF PARTICLES. SCATTERING
CHANGING THE FRAME OF REFERENCE
LAGRANGIAN MECHANICS
CONSERVATION LAWS AND SYMMETRIES IN MANY PARTICLE SYSTEMS
THE RIGID BODY
SMALL OSCILLATIONS
HAMILTON S CANONICAL FORMULATION OF MECHANICS
HAMILTON-JACOBI THEORY
FROM INTEGRABLE TO NON-INTEGRABLE SYSTEMS
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
Titel: A modern approach to classical mechanics
Autor: Iro, Harald
Jahr: 2016
Contents
1 Basic considerations and concepts 1
1.1 Why classical mechanics is still challenging ..............1
1.2 The birth of classical mechanics............................3
1.3 Observations and the resulting pictures ..................5
1.4 Time, space and motion....................................9
1.4.1 Newton s concepts..................................9
1.4.2 The mathematical pictures of space and time . . 11
1.4.3 Kinematics..........................................12
2 Foundations of classical mechanics 15
2.1 Mass, quantity of motion, and force ......................15
2.2 Newton s laws................................................17
2.3 Analytical mechanics........................................19
2.3.1 The basic equations of mechanics..................20
2.3.2 Point masses and forces............................22
2.4 Constants of motion........................................23
2.4.1 Constants of motion and conserved quantities . . 23
2.4.2 Conservation of energy..............................27
2.4.3 Angular momentum and its conservation..........31
3 One-dimensional motion of a particle 35
3.1 Examples of one-dimensional motion......................35
3.2 General features ............................................37
3.3 Back to the examples........................................42
3.3.1 The inclined track..................................42
3.3.2 The plane pendulum................................43
3.3.3 The harmonic oscillator............................46
vii
viii CONTENTS
3.4 The driven, damped oscillator..............................49
3.4.1 The driven oscillator with linear damping .... 50
3.4.2 The periodically driven, damped oscillator .... 52
3.5 Stability of motion..........................................55
3.5.1 Two examples ......................................55
3.5.2 Linear stability analysis............................57
3.6 Anharmonic one-dimensional motion......................63
4 Peculiar motion in two dimensions 71
4.1 The two-dimensional harmonic oscillator..................71
4.2 The Henon-Heiles system ..................................79
4.3 A useless conserved quantity..............................86
4.4 Chaotic behavior............................................89
4.5 Laplace s clock mechanism does not exist................97
5 Motion in a central force 101
5.1 General features of the motion............................102
5.1.1 Conserved quantities................................102
5.1.2 The effective potential..............................105
5.1.3 Properties of the orbits ............................108
5.2 Motion in a 1/r potential..................................110
5.2.1 The case L ^ 0......................................110
5.2.2 Bounded motion for L — 0..........................117
5.3 Motion in the potential V(r) oc l/ra......................118
5.3.1 Mechanical similarity ..............................120
5.4 The Runge-Lenz vector ....................................121
5.5 Integrability vanishes........................................125
5.5.1 The homogeneous magnetic field as the sole force 126
5.5.2 Addition of a central force..........................133
5.5.3 Motion in the symmetry plane....................134
6 Gravitational force between two bodies 139
6.1 Two-body systems..........................................139
6.1.1 Center of mass and relative coordinates..........143
6.1.2 Conserved quantities................................145
6.2 The gravitational interaction ..............................147
6.3 Kepler s laws................................................151
6.3.1 Beyond Kepler s laws ..............................159
CONTENTS ix
6.4 Gravitational potential of large bodies....................160
6.4.1 The potential of a homogeneous sphere ..........161
6.4.2 The potential of an inhomogeneous body .... 164
6.5 On the validity of the gravitational law....................166
7 Collisions of particles. Scattering 171
7.1 Unbounded motion in a central force......................171
7.2 Kinematics of two-particle-collisions......................176
7.2.1 Elastic collisions of two particles..................177
7.2.2 Kinematics of elastic collisions....................179
7.3 Potential scattering..........................................185
7.3.1 The scattering cross section........................186
7.3.2 Scattering in the 1/r potential....................189
8 Changing the frame of reference 195
8.1 Inertial frames..............................................197
8.2 Changing the inertial frame................................198
8.3 Linear transformations of the coordinates................200
8.3.1 Translation of the coordinate system..............200
8.3.2 Rotation of the coordinate system................201
8.4 The Galilean group..........................................205
8.4.1 Transformation of forces............................207
8.5 Transformations to non-inertial frames....................209
8.5.1 Accelerated frames of reference....................209
8.5.2 Rotating frames of reference........................211
8.5.3 Motion in a rotating frame........................216
9 Lagrangian mechanics 223
9.1 Constrained motion ........................................223
9.2 Calculus of variations ......................................231
9.2.1 The Euler-Lagrange equation......................233
9.2.2 Transforming the variables ........................238
9.2.3 Constraints..........................................240
9.3 The Lagrangian..............................................243
9.3.1 Inverse problem in the calculus of variations . . . 243
9.3.2 Inverse problem for Newton s equation of motion 244
9.3.3 The Lagrangian for a single particle..............247
9.3.4 Hamilton s principle................................249
X
CONTENTS
9.3.5 The Lagrangian in generalized coordinates .... 250
9.3.6 Further applications of the Lagrangian...... 255
9.3.7 Nonuniformly moving frames of reference .... 256
10 Conservation laws and symmetries 261
10.1 Equations of motion for N point masses..................261
10.2 The conservation laws......................................265
10.2.1 The motion of the center of mass.........265
10.2.2 Conservation of angular momentum..............267
10.2.3 Conservation of energy..............................271
10.3 The Lagrangian of a system of N particles........275
10.4 Infinitesimal transformations ..............................278
10.4.1 Infinitesimal translations of time..................278
10.4.2 Infinitesimal coordinate transformations.....279
10.4.3 Galilean transformations and constants of motion 282
11 The rigid body 287
11.1 Degrees of freedom of a rigid body............ 288
11.2 Some basics of statics ................... 289
11.2.1 Historical survey....................................289
11.2.2 The basic physical principles......................290
11.2.3 Simple machines....................................292
11.3 Dynamics of the rigid body................................296
11.3.1 Historical landmarks................................296
11.3.2 The motion of a rigid body........................298
11.3.3 The inertia tensor ..................................303
11.3.4 Euler s equations of motion........................309
11.3.5 The motion of a spinning top......................317
11.3.6 The symmetric spinning top........................324
12 Small oscillations 337
12.1 The double pendulum......................................337
12.2 The harmonic approximation..............................341
12.2.1 The general theory..................................341
12.2.2 The double pendulum (again)......................346
12.2.3 Vibrations of a triatomic molecule................349
12.3 From linear chain to vibrating string...........357
12.3.1 The vibrating string................................361
CONTENTS xi
13 Hamiltonian mechanics 365
13.1 Hamilton s equations of motion............................365
13.1.1 A particle in a central force field..................368
13.1.2 The rigid body......................................369
13.1.3 Central force and homogeneous magnetic field . . 371
13.2 Poisson brackets............................................377
13.3 Canonical transformations..................................380
13.3.1 The generating function of a transformation . . . 381
13.3.2 Canonical invariants................................386
13.3.3 Infinitesimal canonical transformations......387
13.4 Symmetries and conservation laws............390
13.5 The flow in phase space....................................392
14 Hamilton-Jacobi theory 397
14.1 Integrability..................................................397
14.1.1 Liouville s theorem on integrability........398
14.1.2 Sketched proof of the theorem ....................400
14.2 Time-independent Hamilton-Jacobi theory........403
14.2.1 The Hamilton-Jacobi equation..........403
14.2.2 Separation of variables..............................405
14.3 The problem of two centers of gravity..........411
15 Three-body systems 419
15.1 The restricted three-body problem............420
15.2 Solutions of the problem....................................426
15.3 Is our planetary system also chaotic?...........434
16 Approximating non-integrable systems 439
16.1 Action-angle variables......................................439
16.1.1 Definition and general properties.........439
16.1.2 Transforming to action and angle variables . . . 443
16.2 Dynamics on the torus......................................452
16.3 Canonical perturbation theory ............................455
16.3.1 The one-dimensional anharmonic oscillator . . . 459
16.3.2 First order corrections..............................462
16.4 The KAM theorem ........................................463
16.5 Is the solar system stable?..................................467
16.5.1 A few historical landmarks............467
xii CONTENTS
16.5.2 On the stability of planetary orbits........ 469
In retrospect 475
Appendix 477
A Coordinates; vector analysis 477
A.l The Euclidean space E3....................................477
A.2 Cartesian coordinates ......................................479
A.3 Orthogonal, curvilinear coordinates........................481
A.3.1 General relations.....................481
A.3.2 Spherical coordinates................................483
A.3.3 Cylindrical coordinates ............................485
B Rotations and tensors 487
B.l Rotations.......................... 487
B.2 Tensors ........................... 490
C Green s functions 493
C.l The Dirac Afunction........................................493
C.l.l Distributions........................................493
C.l.2 The S-function......................................494
C.2 Fourier transforms..........................................495
C.3 Linear differential equations................................497
C.3.1 The Green s function................................497
C.3.2 The equation of the damped oscillator......498
Bibliography 503
Index 509
|
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| spelling | Iro, Harald Verfasser (DE-588)1103144650 aut A modern approach to classical mechanics Harald Iro, retired from Institute for Theoretical Physics, Johannes Kepler University Linz, Austria Second edition New Jersey World Scientific [2016] xii, 515 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Mechanics Statistical mechanics Mechanik (DE-588)4038168-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Mechanik (DE-588)4038168-7 s DE-604 LoC Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028371735&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028371735&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iro, Harald A modern approach to classical mechanics Mechanics Statistical mechanics Mechanik (DE-588)4038168-7 gnd |
| subject_GND | (DE-588)4038168-7 (DE-588)4123623-3 |
| title | A modern approach to classical mechanics |
| title_auth | A modern approach to classical mechanics |
| title_exact_search | A modern approach to classical mechanics |
| title_full | A modern approach to classical mechanics Harald Iro, retired from Institute for Theoretical Physics, Johannes Kepler University Linz, Austria |
| title_fullStr | A modern approach to classical mechanics Harald Iro, retired from Institute for Theoretical Physics, Johannes Kepler University Linz, Austria |
| title_full_unstemmed | A modern approach to classical mechanics Harald Iro, retired from Institute for Theoretical Physics, Johannes Kepler University Linz, Austria |
| title_short | A modern approach to classical mechanics |
| title_sort | a modern approach to classical mechanics |
| topic | Mechanics Statistical mechanics Mechanik (DE-588)4038168-7 gnd |
| topic_facet | Mechanics Statistical mechanics Mechanik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028371735&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028371735&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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