An introduction to Lagrangian mechanics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey
World Scientific
2015
|
| Ausgabe: | 2nd edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references (pages 301-302) and index |
| Beschreibung: | xviii, 305 pages illustrations |
| ISBN: | 9789814623612 981462361X 9789814623629 9814623628 |
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| 100 | 1 | |a Brizard, Alain Jean |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An introduction to Lagrangian mechanics |c Alain J. Brizard, Saint Michael's College, USA |
| 250 | |a 2nd edition | ||
| 264 | 1 | |a New Jersey |b World Scientific |c 2015 | |
| 300 | |a xviii, 305 pages |b illustrations | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (pages 301-302) and index | ||
| 650 | 4 | |a Lagrangian functions |v Textbooks | |
| 650 | 4 | |a Hamiltonian systems |v Textbooks | |
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Datensatz im Suchindex
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|---|---|
| adam_text | An Introduction
to Lagrangian Mechanics
2nd Edition
An Introduction to Lagrangian Mechanics begins with a proper historical
perspective on the Lagrangian method by presenting Fermat’s Principle of
Least Time (as an introduction to the Calculus of Variations) as well as the
principles of Maupertuis, Jacobi, and d’Alembert that preceded Hamilton’s
formulation of the Principle of Least Action, from which the Euler-Lagrange
equations of motion are derived. Other additional topics not traditionally
presented in undergraduate textbooks include the treatment of constraint
forces in Lagrangian Mechanics; Routh’s procedure for Lagrangian systems
with symmetries; the art of numerical analysis for physical systems; variational
formulations for several continuous Lagrangian systems; an introduction to
elliptic functions with applications in Classical Mechanics; and Noncanonical
Hamiltonian Mechanics and perturbation theory.
The Second Edition includes a larger selection of examples and problems
(with hints) in each chapter and continues the strong emphasis of the First
Edition on the development and application of mathematical methods (mostly
calculus) to the solution of problems in Classical Mechanics.
New material has been added to most chapters. For example, a new derivation
of the Noether theorem for discrete Lagrangian systems is given and a
modified Rutherford scattering problem is solved exactly to show that the
total scattering cross section associated with a confined potential (i.e., which
vanishes beyond a certain radius) yields the hard-sphere result. The Frenet-
Serret formulas for the Coriolis-corrected projectile motion are presented,
where the Frenet-Serret torsion is shown to be directly related to the Coriolis
deflection, and a new treatment of the sleeping-top problem is given.
Contents^
Preface to the Second Edition vii
Preface to the First Edition ix
1. The Calculus of Variations 1
11 Foundations of the Calculus of Variations................. 1
1 1 1 A Simple Minimization Problem ..................... 1
11.2 Methods of the Calculus of Variations.......... 3
1.1.3 Path of Shortest Distance and Geodesic Equation 8
1.2 Classical Variational Problems .......................... 10
1.2.1 Isoperimetric Problem.......................... 11
1.2.2 Brachistochrone Problem........................ 13
1.3 Fermat’s Principle of Least Time......................... 15
1.3.1 Light Propagation in a Nonuniform Medium ... 17
1.3.2 Snell’s Law.................................... 19
1.3.3 Application of Fermat’s Principle.............. 20
1.4 Geometric Formulation of Ray Optics*..................... 21
1-4.1 General Geometric Optics ......................... 21
1.4.2 Light-ray Frenet-Serret Equations.............. 23
1.4.3 Light Propagation in Spherical Geometry........ 24
1.4.4 Geodesic Representation of Light Propagation . . 26
1.4.5 Wavefront Representation ......................... 29
L5 Summary.................................................. 31
L6 Problems................................................. 31
2. Lagrangian Mechanics 37
2-1 Maupertuis-Jacobi Principle of Least Action.............. 38
xiii
xiv An Introduction to Lagrangian Mechanics
2.1.1 Maupertuis’ Principle......................... 38
2.1.2 Jacobi’s Principle............................ 39
2.2 d’Alembert’s Principle................................. 41
2.2.1 Principle of Virtual Work..................... 41
2.2.2 Lagrange’s Equations from d’Alembert’s Principle 42
2.3 Hamilton’s Principle................................... 45
2.3.1 Constraint Forces............................. 45
2.3.2 Generalized Coordinates in Configuration Space . 47
2.3.3 Constrained Motion on a Surface ................ 48
2.3.4 Euler-Lagrange Equations ....................... 50
2.3.5 Four-step Lagrangian Method................... 51
2.3.6 Lagrangian Mechanics in Curvilinear Coordinates* 52
2.4 Lagrangian Mechanics in Configuration Space............ 53
2.4.1 Example I: Pendulum........................... 53
2.4.2 Example II: Bead on a Rotating Hoop ............ 54
2.4.3 Example III: Rotating Pendulum ................. 56
2.4.4 Example IV: Compound Atwood Machine .... 57
2.4.5 Example V: Pendulum with Oscillating Fulcrum . 59
2.5 Symmetries and Conservation Laws....................... 61
2.5.1 Energy Conservation Law....................... 62
2.5.2 Momentum Conservation Laws.................... 62
2.5.3 Invariance Properties of a Lagrangian......... 63
2.5.4 Lagrangian Mechanics with Symmetries.......... 64
2.5.5 Routh’s Procedure............................. 65
2.6 Lagrangian Mechanics in the CM Frame................... 67
2.7 Summary................................................ 69
2.8 Problems............................................... 70
3. Hamiltonian Mechanics 77
3.1 Hamilton’s Canonical Equations ........................ 77
3.2 Legendre Transformation*............................... 79
3.3 Hamiltonian Optics and Wave-Particle Duality*.......... ^0
3.4 Motion in an Electromagnetic Field .................... ^2
3.4.1 Euler-Lagrange Equations ....................... ^2
3.4.2 Gauge Invariance.............................. ^4
3.4.3 Canonical Hamilton’s Equations................ ^4
3.4.4 Maupertuis’ Principle for Particle-Beam Optics . 85
3.5 One-degree-of-freedom Hamiltonian Dynamics............. ^6
3.5.1 Energy Method .................................. 86
Contents
XV
3.5.2 Simple Harmonic Oscillator.................
3.5.3 Morse Potential............................
3.5.4 Pendulum ..................................
3.5.5 Constrained Motion on the Surface of a Cone . .
3.6 Summary............................................
3.7 Problems .................
88
89
93
98
100
100
4. Motion in a Central-Force Field
4.1 Motion in a Central-Force Field....................
4.1.1 Lagrangian Formalism ......................
4.1.2 Hamiltonian Formalism......................
4.2 Homogeneous Central Potentials*
4.2.1 The Virial Theorem.........................
4.2.2 General Properties of Homogeneous Potentials . .
4.3 Kepler Problem.....................................
4.3.1 Bounded Keplerian Orbits
4.3.2 Unbounded Kepler ian Orbits
4.3.3 Laplace-Runge-Lenz Vector*
4.4 Isotropic Simple Harmonic Oscillator
4.5 Internal Reflection inside a Well
4.6 Summary..............................
4.7 Problems ................
105
105
106
109
110
110
112
113
114
117
118
121
122
124
125
5· Collisions and Scattering Theory
5.1 Two-Particle Collisions in the LAB Frame
5.2 Two-Particle Collisions in the CM Frame
5.3 Connection between the CM and LAB Frames
5.4 Scattering Cross Sections .... rrames
5.4.1 Definitions .
„ f,4- Sections in CM and LAB FVames
}.5 Hard-Sphere Scattering
5.6 Rutherford Scattering
5.6.1 Classical Rutherford Scattering....
5.6.2 Modified Rutherford Scattering
5.7 Soft-Sphere Scattering
5.8 Elastic Scattering by a Hard Surface
5.9 Summary..............
5.10 Problems ..............
133
134
137
138
139
140
141
143
145
145
147
150
152
154
155
XVI
An Introduction to Lagrangian Mechanics
6. Motion in a Non-Inertial Frame 161
6.1 Time Derivatives in Rotating Frames..................... 161
6.2 Accelerations in Rotating Frames........................ 163
6.3 Lagrangian Formulation of Non-Inertial Motion........... 164
6.4 Motion Relative to Earth................................ 166
6.4.1 Coriolis-corrected Projectile Motion............ 168
6.4.2 Frenet-Serret-Coriolis Formulas................. 170
6.4.3 Free-Fall Problem Revisited..................... 172
6.4.4 Foucault Pendulum............................. 172
6.5 Summary................................................. 175
6.6 Problems................................................ 176
7. Rigid Body Motion 181
7.1 Inertia Tensor of a Rigid Body........................... 181
7.1.1 Discrete Particle Distribution .................. 181
7.1.2 Parallel-Axes Theorem......................... 183
7.1.3 Continuous Particle Distribution.............. 184
7.1.4 Principal Axes of Inertia..................... 185
7.2 Eulerian Rigid-Body Dynamics............................. 189
7.2.1 Euler Equations............................... 189
7.2.2 Euler Equations for a Torque-free Symmetric Top 190
7.2.3 Euler Equations for a Free Asymmetric Top . . . 194
7.3 Lagrangian Rigid-Body Dynamics .......................... 197
7.3.1 Eulerian Angles as Generalized Coordinates ... 197
7.3.2 Angular Velocity in Terms of Eulerian Angles . . 198
7.3.3 Rotational Kinetic Energy of a Symmetric Top . . 199
7.3.4 Space-frame Precession and Space-cone Solutions 202
7.4 Symmetric Top with One Fixed Point....................... 204
7.4.1 Nutation...................................... 206
7.4.2 Slow and Fast Precession...................... 209
7.4.3 The Sleeping Top.............................. 209
7.5 Summary.................................................. 212
7.6 Problems................................................. 213
8. Normal-Mode Analysis 219
8.1 Stability of Equilibrium Points.......................... 219
8.1.1 Bead on a Rotating Hoop....................... 219
8.1.2 Circular Orbits in Central-Force Fields....... 220
Contents xvii
8.2 Small Oscillations about Stable Equilibria...............221
8.3 Normal-Mode Analysis of Coupled Oscillations.............223
8.3.1 Normal-Mode Analysis..............................223
8.3.2 Coupled Simple Harmonic Oscillators...............226
8.3.3 Coupled Nonlinear Oscillators.................... 228
8.3.4 Stability of the Sleeping Top.................... 231
8.4 Summary................................................. 235
8.5 Problems................................................ 235
9. Continuous Lagrangian Systems 241
9.1 Waves on a Stretched String..............................241
9.1.1 Wave Equation ....................................241
9.1.2 Lagrangian Formulation............................242
9.2 Variational Principle for Field Theory*................. 243
9.2.1 Lagrangian Formulation............................244
9.2.2 Noether Method and Conservation Laws..............245
9.3 Schroedinger’s Equation................................. 248
9.4 Euler Equations for a Perfect Fluid......................250
9.4.1 Lagrangian Formulation............................251
9.4.2 Energy-Momentum Conservation Laws.................252
9.5 Summary..................................................253
9.6 Problems.................................................254
Appendix A Basic Mathematical Methods 255
A.l Roots of a General Cubic Polynomial......................255
A.2 Integration by Trigonometric Substitution................257
A.2.1 Trigonometric Functions.......................... 257
A.2.2 Hyperbolic-Trigonometric Functions................258
A.3 Frenet-Serret Formulas.................................. 259
A.3.1 Frenet Frame......................................259
A.3.2 Darboux Frame.....................................261
A.3.3 Example: Seiffert Spiral on the Unit Sphere . . . 262
A.3.4 Frenet-Serret Formulas for Helical Path...........264
A.4 Linear Algebra.......................................... 265
A.4.1 Matrix Algebra................................... 266
A.4.2 Eigenvalue Analysis of a 2 x 2 Matrix.............268
A.5 Numerical Analysis.......................................272
Appendix В Elliptic Functions and Integrals*
275
xviii An Introduction to Lagrangian Mechanics
B.l Jacobi Elliptic Functions................................. 275
B.1.1 Definitions and Notation............................ 275
ВЛ.2 Motion in a Quartic Potential.................... 279
B.2 Weierstrass Elliptic Functions............................ 280
B.2.1 Definitions and Notation............................ 280
B.2.2 Motion in a Cubic Potential...................... 285
B. 3 Connection between Elliptic Functions..................... 287
Appendix C Noncanonical Hamiltonian Mechanics* 291
C. l Differential Geometry..................................... 291
C.2 Lagrange and Poisson Tensors.............................. 293
C.3 Hamiltonian Perturbation Theory........................... 295
Bibliography 301
Index 303
|
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| spelling | Brizard, Alain Jean Verfasser aut An introduction to Lagrangian mechanics Alain J. Brizard, Saint Michael's College, USA 2nd edition New Jersey World Scientific 2015 xviii, 305 pages illustrations txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (pages 301-302) and index Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd rswk-swf Lagrange-Formalismus (DE-588)4316154-6 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027976285&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027976285&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Brizard, Alain Jean An introduction to Lagrangian mechanics Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd |
| subject_GND | (DE-588)4316154-6 |
| title | An introduction to Lagrangian mechanics |
| title_auth | An introduction to Lagrangian mechanics |
| title_exact_search | An introduction to Lagrangian mechanics |
| title_full | An introduction to Lagrangian mechanics Alain J. Brizard, Saint Michael's College, USA |
| title_fullStr | An introduction to Lagrangian mechanics Alain J. Brizard, Saint Michael's College, USA |
| title_full_unstemmed | An introduction to Lagrangian mechanics Alain J. Brizard, Saint Michael's College, USA |
| title_short | An introduction to Lagrangian mechanics |
| title_sort | an introduction to lagrangian mechanics |
| topic | Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus (DE-588)4316154-6 gnd |
| topic_facet | Lagrangian functions Textbooks Hamiltonian systems Textbooks Lagrange-Formalismus |
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