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9780521631761

Classical Dynamics : A Contemporary Approach

by
  • ISBN13:

    9780521631761

  • ISBN10:

    0521631769

  • Format: Hardcover
  • Copyright: 1998-08-13
  • Publisher: Cambridge University Press
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Summary

Recent advances in the study of dynamical systems have revolutionized the way that classical mechanics is taught and understood. This new and comprehensive textbook provides a complete description of this fundamental branch of physics. The authors cover all the material that one would expect to find in a standard graduate course: Lagrangian and Hamiltonian dynamics, canonical transformations, the Hamilton-Jacobi equation, perturbation methods, and rigid bodies. They also deal with more advanced topics such as the relativistic Kepler problem, Liouville and Darboux theorems, and inverse and chaotic scattering. A key feature of the book is the early introduction of geometric (differential manifold) ideas, as well as detailed treatment of topics in nonlinear dynamics (such as the KAM theorem) and continuum dynamics (including solitons). Over 200 homework exercises are included. It will be an ideal textbook for graduate students of physics, applied mathematics, theoretical chemistry, and engineering, as well as a useful reference for researchers in these fields. A solutions manual is available exclusively for instructors.

Table of Contents

List of Worked Examples xix(2)
Preface xxi
Two Paths Through the Book xxiv
1 FUNDAMENTALS OF MECHANICS
1(47)
1.1 Elementary Kinematics
1(4)
1.1.1 Trajectories of Point Particles
1(2)
1.1.2 Position, Velocity, and Acceleration
3(2)
1.2 Principles of Dynamics
5(8)
1.2.1 Newton's Laws
5(1)
1.2.2 The Two Principles
6(1)
Principle 1
7(1)
Principle 2
7(2)
Discussion
9(1)
1.2.3 Consequences of Newton's Equations
10(1)
Introduction
10(1)
Force is a Vector
11(2)
1.3 One-Particle Dynamical Variables
13(9)
1.3.1 Momentum
14(1)
1.3.2 Angular Momentum
14(1)
1.3.3 Energy and Work
15(1)
In Three Dimensions
15(3)
Application to One-Dimensional Motion
18(4)
1.4 Many-Particle Systems
22(7)
1.4.1 Momentum and Center of Mass
22(1)
Center of Mass
22(2)
Momentum
24(1)
Variable Mass
24(2)
1.4.2 Energy
26(1)
1.4.3 Angular Momentum
27(2)
1.5 Examples
29(13)
1.5.1 Velocity Phase Space and Phase Portraits
29(1)
The Cosine Potential
29(2)
The Kepler Problem
31(2)
1.5.2 A System with Energy Loss
33(5)
1.5.3 Noninertial Frames and the Equivalence Principle
38(1)
Equivalence Principle
38(3)
Rotating Frames
41(1)
Problems
42(6)
2 LAGRANGIAN FORMULATION OF MECHANICS
48(60)
2.1 Constraints and Configuration Manifolds
49(13)
2.1.1 Constraints
49(1)
Constraint Equations
49(1)
Constraints and Work
50(4)
2.1.2 Generalized Coordinates
54(3)
2.1.3 Examples of Configuration Manifolds
57(1)
The Finite Line
57(1)
The Circle
57(1)
The Plane
57(1)
The Two-Sphere S(2)
57(3)
The Double Pendulum
60(1)
Discussion
60(2)
2.2 Lagrange's Equations
62(15)
2.2.1 Derivation of Lagrange's Equations
62(5)
2.2.2 Transformation of Lagrangians
67(1)
Equivalent Lagrangians
67(1)
Coordinate Independence
68(1)
Hessian Condition
69(1)
2.2.3 Conservation of Energy
70(2)
2.2.4 Charged Particle in an Electromagnetic Field
72(1)
The Lagrangian
72(2)
A Time-Dependent Coordinate Transformation
74(3)
2.3 Central Force Motion
77(15)
2.3.1 The General Central Force Problem
77(1)
Statement of the Problem; Reduced Mass
77(1)
Reduction to Two Freedoms
78(1)
The Equivalent One-Dimensional Problem
79(5)
2.3.2 The Kepler Problem
84(4)
2.3.3 Bertrand's Theorem
88(4)
2.4 The Tangent Bundle TQ
92(11)
2.4.1 Dynamics on TQ
92(1)
Velocities Do Not Lie in Q
92(1)
Tangent Spaces and the Tangent Bundle
93(2)
Lagrange's Equations and Trajectories on TQ
95(2)
2.4.2 TQ as a Differential Manifold
97(1)
Differential Manifolds
97(3)
Tangent Spaces and Tangent Bundles
100(2)
Application to Lagrange's Equations
102(1)
Problems
103(5)
3 TOPICS IN LAGRANGIAN DYNAMICS
108(39)
3.1 The Variational Principle and Lagrange's Equations
108(10)
3.1.1 Derivation
108(1)
The Action
108(2)
Hamilton's Principle
110(2)
Discussion
112(2)
3.1.2 Inclusion of Constraints
114(4)
3.2 Symmetry and Conservation
118(10)
3.2.1 Cyclic Coordinates
118(1)
Invariant Submanifolds and Conservation of Momentum
118(1)
Transformations, Passive and Active
119(4)
Three Examples
123(1)
3.2.2 Noether's Theorem
124(1)
Point Transformations
124(1)
The Theorem
125(3)
3.3 Nonpotential Forces
128(6)
3.3.1 Dissipative Forces in the Lagrangian Formalism
129(1)
Rewriting the EL Equations
129(1)
The Dissipative and Rayleigh Functions
129(2)
3.3.2 The Damped Harmonic Oscillator
131(3)
3.3.3 Comment on Time-Dependent Forces
134(1)
3.4 A Digression on Geometry
134(9)
3.4.1 Some Geometry
134(1)
Vector Fields
134(1)
One-Forms
135(1)
The Lie Derivative
136(2)
3.4.2 The Euler-Lagrange Equations
138(1)
3.4.3 Noether's Theorem
139(1)
One-Parameter Groups
139(1)
The Theorem
140(3)
Problems
143(4)
4 SCATTERING AND LINEAR OSCILLATIONS
147(54)
4.1 Scattering
147(31)
4.1.1 Scattering by Central Forces
147(1)
General Considerations
147(6)
The Rutherford Cross Section
153(1)
4.1.2 The Inverse Scattering Problem
154(1)
General Treatment
154(2)
Example: Coulomb Scattering
156(1)
4.1.3 Chaotic Scattering, Cantor Sets, and Fractal Dimension
157(1)
Two Disks
158(4)
Three Disks, Cantor Sets
162(4)
Fractal Dimension and Lyapunov Exponent
166(3)
Some Further Results
169(1)
4.1.4 Scattering of a Charge by a Magnetic Dipole
170(1)
The Stormer Problem
170(1)
The Equatorial Limit
171(3)
The General Case
174(4)
4.2 Linear Oscillations
178(19)
4.2.1 Linear Approximation: Small Vibrations
178(1)
Linearization
178(2)
Normal Modes
180(3)
4.2.2 Commensurate and Incommensurate Frequencies
183(1)
The Invariant Torus T
183(2)
The Poincare Map
185(2)
4.2.3 A Chain of Coupled Oscillators
187(1)
General Solution
187(2)
The Finite Chain
189(3)
4.2.4 Forced and Damped Oscillators
192(1)
Forced Undamped Oscillator
192(1)
Forced Damped Oscillator
193(4)
Problems
197(4)
5 HAMILTONIAN FORMULATION OF MECHANICS
201(83)
5.1 Hamilton's Canonical Equations
202(22)
5.1.1 Local Considerations
202(1)
From the Lagrangian to the Hamiltonian
202(5)
A Brief Review of Special Relativity
207(4)
The Relativistic Kepler Problem
211(1)
5.1.2 The Legendre Transform
212(3)
5.1.3 Unified Coordinates on T*Q and Poisson Brackets
215(1)
The Xi Notation
215(2)
Variational Derivation of Hamilton's Equations
217(1)
Poisson Brackets
218(4)
Poisson Brackets and Hamiltonian Dynamics
222(2)
5.2 Symplectic Geometry
224(7)
5.2.1 The Cotangent Manifold
224(1)
5.2.2 Two-Forms
225(1)
5.2.3 The Symplectic Form w
226(5)
5.3 Canonical Transformations
231(22)
5.3.1 Local Considerations
231(1)
Reduction on T*Q by Constants of the Motion
231(1)
Definition of Canonical Transformations
232(2)
Changes Induced by Canonical Transformations
234(2)
Two Examples
236(3)
5.3.2 Intrinsic Approach
239(1)
5.3.3 Generating Functions of Canonical Transformations
240(1)
Generating Functions
240(2)
The Generating Functions Gives the New
Hamiltonian
242(2)
Generating Functions of Type
244(4)
5.3.4 One-Parameter Groups of Canonical Transformations
248(1)
Infinitesimal Generators of One-Parameter Groups;
Hamiltonian Flows
249(2)
The Hamiltonian Noether Theorem
251(1)
Flows and Poisson Brackets
252(1)
5.4 Two Theorems: Liouville and Darboux
253(22)
5.4.1 Liouville's Volume Theorem
253(1)
Volume
253(4)
Integration on T*Q; The Liouville Theorem
257(3)
Poincare Invariants
260(1)
Density of States
261(5)
5.4.2 Darboux's Theorem
266(3)
The Theorem
269(1)
Reduction
270(5)
Problems
275(5)
Canonicity Implies PB Preservation
280(4)
6 TOPICS IN HAMILTONIAN DYNAMICS
284(98)
6.1 The Hamilton-Jacobi Method
284(23)
6.1.1 The Hamilton-Jacobi Equation
285(1)
Derivation
285(1)
Properties of Solutions
286(2)
Relation to the Action
288(2)
6.1.2 Separation of Variables
290(1)
The Method of Separation
291(3)
Example: Charged Particle in a Magnetic Field
294(7)
6.1.3 Geometry and the HJ Equation
301(2)
6.1.4 The Analogy Between Optics and the HJ Method
303(4)
6.2 Completely Integrable Systems
307(25)
6.2.1 Action-Angle Variables
307(1)
Invariant Tori
307(2)
The Phi(Alpha) and J(Alpha)
309(2)
The Canonical Transformation to AA Variables
311(3)
Example: A Particle on a Vertical Cylinder
314(6)
6.2.2 Liouville's Integrability Theorem
320(1)
Complete Integrability
320(1)
The Tori
321(2)
The J(Alpha)
323(1)
Example: the Neumann Problem
324(4)
6.2.3 Motion on the Tori
328(1)
Rational and Irrational Winding Lines
328(3)
Fourier Series
331(1)
6.3 Perturbation Theory
332(27)
6.3.1 Example: The Quartic Oscillator; Secular Perturbation Theory
332(4)
6.3.2 Hamiltonian Perturbation Theory
336(1)
Perturbation via Canonical Transformations
337(2)
Averaging
339(1)
Canonical Perturbation Theory in One Freedom
340(6)
Canonical Perturbation Theory in Many Freedoms
346(5)
The Lie Transformation Method
351(6)
Example: The Quartic Oscillator
357(2)
6.4 Adiabatic Invariance
359(18)
6.4.1 The Adiabatic Theorem
360(1)
Oscillator with Time-Dependent Frequency
360(1)
The Theorem
361(2)
Remarks on N greater than 1
363(1)
6.4.2 Higher Approximations
364(1)
6.4.3 The Hannay Angle
365(6)
6.4.4 Motion of a Charged Particle in a Magnetic Field
371(1)
The Action Integral
371(3)
Three Magnetic Adiabatic Invariants
374(3)
Problems
377(5)
7 NONLINEAR DYNAMICS
382(110)
7.1 Nonlinear Oscillators
383(13)
7.1.1 A Model System
383(3)
7.1.2 Driven Quartic Oscillator
386(1)
Damped Driven Quartic Oscillator; Harmonic Analysis
387(3)
Undamped Driven Quartic Oscillator
390(1)
7.1.3 Example: The van der Pol Oscillator
391(5)
7.2 Stability of Solutions
396(22)
7.2.1 Stability of Autonomous Systems
397(1)
Definitions
397(2)
The Poincare-Bendixon Theorem
399(1)
Linearization
400(10)
7.2.2 Stability of Nonautonomous Systems
410(1)
The Poincare Map
410(3)
Linearization of Discrete Maps
413(4)
Example: The Linearized Henon Map
417(1)
7.3 Parametric Oscillators
418(13)
7.3.1 Floquet Theory
419(1)
The Floquet Operator R
419(1)
Standard Basis
420(1)
Eigenvalues of R and Stability
421(3)
Dependence on G
424(1)
7.3.2 The Vertically Driven Pendulum
424(1)
The Mathieu Equation
424(2)
Stability of the Pendulum
426(1)
The Inverted Pendulum
427(2)
Damping
429(2)
7.4 Discrete Maps; Chaos
431(21)
7.4.1 The Logistic Map
431(1)
Definition
432(1)
Fixed Points
432(2)
Period Doubling
434(8)
Universality
442(2)
Further Remarks
444(1)
7.4.2 The Circle Map
445(1)
The Damped Driven Pendulum
445(1)
The Standard Sine Circle Map
446(1)
Rotation Number and the Devil's Staircase
447(3)
Fixed Points of the Circle Map
450(2)
7.5 Chaos in Hamiltonian Systems and the KAM Theorem
452(31)
7.5.1 The Kicked Rotator
453(1)
The Dynamical System
453(1)
The Standard Map
454(1)
Poincare Map of the Perturbed System
455(5)
7.5.2 The Henon Map
460(3)
7.5.3 Chaos in Hamiltonian Systems
463(1)
Poincare-Birkhoff Theorem
464(2)
The Twist Map
466(1)
Numbers and Properties of the Fixed Points
467(1)
The Homoclinic Tangle
468(4)
The Transition to Chaos
472(2)
7.5.4 The KAM Theorem
474(1)
Background
474(1)
Two Conditions: Hessian and Diophantine
475(2)
The Theorem
477(3)
A Brief Description of the Proof of KAM
480(3)
Problems
483(3)
Number Theory
486(6)
The Unit Interval
486(1)
A Diophantine Condition
487(1)
The Circle and the Plane
488(1)
KAM and Continued Fractions
489(3)
8 RIGID BODIES
492(61)
8.1 Introduction
492(18)
8.1.1 Rigidity and Kinematics
492(1)
Definition
492(1)
The Angular Velocity Vector Omega
493(2)
8.1.2 Kinetic Energy and Angular Momentum
495(1)
Kinetic Energy
495(3)
Angular Momentum
498(1)
8.1.3 Dynamics
499(1)
Space and Body Systems
499(1)
Dynamical Equations
500(3)
Example: The Gyrocompass
503(2)
Motion of the Angular Momentum J
505(1)
Fixed Points and Stability
506(2)
The Poinsot Construction
508(2)
8.2 The Lagrangian and Hamiltonian Formulations
510(16)
8.2.1 The Configuration Manifold Q(R)
510(1)
Inertial, Space, and Body Systems
510(1)
The Dimension of Q(R)
511(1)
The Structure of Q(R)
512(2)
8.2.2 The Lagrangian
514(1)
Kinetic Energy
514(1)
The Constraints
515(1)
8.2.3 The Euler-Lagrange Equations
516(1)
Derivation
516(2)
The Angular Velocity Matrix Omega
518(1)
8.2.4 The Hamiltonian Formalism
519(1)
8.2.5 Equivalence to Euler's Equations
520(1)
Antisymmetric Matrix-Vector Correspondence
520(1)
The Torque
521(1)
The Angular Velocity Pseudovector and Kinematics
522(1)
Transformations of Velocities
523(1)
Hamilton's Canonical Equations
524(1)
8.2.6 Discussion
525(1)
8.3 Euler Angles and Spinning Tops
526(17)
8.3.1 Euler Angles
526(1)
Definition
526(1)
R in Terms of the Euler Angles
527(2)
Angular Velocities
529(2)
Discussion
531(2)
8.3.2 Geometric Phase for a Rigid Body
533(2)
8.3.3 Spinning Tops
535(1)
The Lagrangian and Hamiltonian
536(1)
The Motion of the Top
537(2)
Nutation and Precession
539(3)
Quadratic Potential; the Neumann Problem
542(1)
8.4 Cayley-Klein Parameters
543(6)
8.4.1 2 x 2 Matrix Representation of 3-Vectors and Rotations
543(1)
3-Vectors
543(1)
Rotations
544(1)
8.4.2 The Pauli Matrices and CK Parameters
544(1)
Definitions
544(1)
Finding R(U)
545(1)
Axis and Angle in terms of the CK Parameters
546(1)
8.4.3 Relation Between SU(2) and SO(3)
547(2)
Problems
549(4)
9 CONTINUUM DYNAMICS
553(95)
9.1 Lagrangian Formulation of Continuum Dynamics
553(12)
9.1.1 Passing to the Continuum Limit
553(1)
The Sine-Gordon Equation
553(3)
The Wave and Klein-Gordon Equations
556(1)
9.1.2 The Variational Principle
557(1)
Introduction
557(1)
Variational Derivation of the EL Equations
557(3)
The Functional Derivative
560(1)
Discussion
560(1)
9.1.3 Maxwell's Equations
561(1)
Some Special Relativity
561(1)
Electromagnetic Fields
562(2)
The Lagrangian and the EL Equations
564(1)
9.2 Noether's Theorem and Relativistic Fields
565(18)
9.2.1 Noether's Theorem
565(1)
The Theorem
565(1)
Conserved Currents
566(1)
Energy and Momentum in the Field
567(2)
Example: The Electromagnetic Energy-Momentum Tensor
569(2)
9.2.2 Relativistic Fields
571(1)
Lorentz Transformations
571(1)
Lorentz Invariant XXX and Conservation
572(4)
Free Klein-Gordon Fields
576(1)
Complex K-G Field and Interaction with the Maxwell Field
577(2)
Discussion of the Coupled Field Equations
579(1)
9.2.3 Spinors
580(1)
Spinor Fields
580(2)
A Spinor Field Equation
582(1)
9.3 The Hamiltonian Formalism
583(11)
9.3.1 The Hamiltonian Formalism for Fields
583(1)
Definitions
583(1)
The Canonical Equations
584(2)
Poisson Brackets
586(2)
9.3.2 Expansion in Orthonormal Functions
588(1)
Orthonormal Functions
589(1)
Particle-like Equations
590(1)
Example: Klein-Gordon
591(3)
9.4 Nonlinear Field Theory
594(16)
9.4.1 The Sine-Gordon Equation
594(1)
Soliton Solutions
595(2)
Properties of s(G) Solitions
597(2)
Multiple-Soliton Solutions
599(2)
Generating Soliton Solutions
601(4)
Nonsoliton Solutions
605(3)
Josephson Junctions
608(1)
9.4.2 The Nonlinear K-G Equation
608(1)
The Lagrangian and the EL Equation
608(1)
Kinks
609(1)
9.5 Fluid Dynamics
610(22)
9.5.1 The Euler and Navier-Stokes Equations
611(1)
Substantial Derivative and Mass Conservation
611(1)
Euler's Equation
612(2)
Viscosity and Incompressibility
614(1)
The Navier-Stokes Equations
615(1)
Turbulence
616(2)
9.5.2 The Burgers Equation
618(1)
The Equation
618(2)
Asymptotic Solution
620(2)
9.5.3 Surface Waves
622(1)
Equations for the Waves
622(2)
Linear Gravity Waves
624(2)
Nonlinear Shallow Water Waves: the KdV Equation
626(3)
Single KdV Solitons
629(2)
Multiple KdV Solitons
631(1)
9.6 Hamiltonian Formalism for Nonlinear Field Theory
632(14)
9.6.1 The Field Theory Analog of Particle Dynamics
633(1)
From Particles to Fields
633(1)
Dynamical Variables and Equations of Motion
634(1)
9.6.2 The Hamiltonian Formalism
634(1)
The Gradient
634(2)
The Symplectic Form
636(1)
The Condition for Canonicity
636(1)
Poisson Brackets
636(1)
9.6.3 The kdV Equation
637(1)
KdV as a Hamiltonian Field
637(1)
Constants of the Motion
638(1)
Generating the Constants of the Motion
639(1)
More on Constants of the Motion
640(2)
9.6.4 The Sine-Gordon Equation
642(1)
Two-Component Field Variables
642(1)
s(G) as a Hamiltonian Field
643(3)
Problems
646(2)
EPILOGUE 648(1)
APPENDIX: VECTOR SPACES 649(7)
General Vector Spaces 649(2)
Linear Operators 651(1)
Inverses and Eigenvalues 652(1)
Inner Products and Hermitian Operators 653(3)
BIBLIOGRAPHY 656(7)
INDEX 663

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