✍ Scribed by Harald J W Muller-Kirsten
No coin nor oath required. For personal study only.
The text covers the entire domain of basic classical mechanics and relativity theory (special and general) and has been revised mainly for the purpose of adding exercises without worked solutions that were missing in the first edition. To retain the format of a readable, yet advanced introductory text that can serve as the companion text for a course in mechanics, the more than 100 new exercises on diverse topics are of moderate range; answers are given and occasionally hints are provided. As before, the text aims to cover the entire spectrum of theoretical mechanics from Newton to Einstein. The reader can observe how in the course of time, deeper and deeper insights were achieved with the development of the basic equations of Newton to those of Euler and Lagrange, and to the geodesic equations of space-time and Einstein's relativity. To include diverse problems, a small section on this topic has been added.
Contents
Preface to Second Edition
Preface to First Edition
1. Introduction
1.1 Introduction
2. Recapitulation of Newtonian Mechanics
2.1 Introductory Remarks
2.2 Recapitulation of Newton’s Laws
2.3 Further Definitions and Rotational Motion
2.4 Conservative Forces
2.5 Mechanics of a System of Particles
2.6 Newton’s Law of Gravitation
2.7 Friction
2.8 Miscellaneous Examples
2.9 Problems without Worked Solutions
3. The Lagrange Formalism
3.1 Introductory Remarks
3.2 The Generalized Coordinates
3.3 The Principle of Virtual Work
3.4 D’Alembert’s Principle, Lagrange Equations
3.5 Hamilton’s Variational Principle, and Euler–Lagrange Equations
3.5.1 Hamilton’s variational principle
3.5.2 Hamilton’s principle for conservative systems
3.5.3 Hamilton’s principle for holonomic systems
3.5.4 Hamilton’s principle for nonholonomic systems
3.5.5 The general procedure
3.6 Symmetry Properties and Conservation Laws
3.7 Miscellaneous Examples
3.8 Problems without Worked Solutions
4. The Canonical or Hamilton Formalism
4.1 Introductory Remarks
4.2 Hamilton’s Equations of Motion
4.3 Physical Significance of the Hamilton Function
4.4 Variational Principle for Hamilton’s Equations
4.5 Transformation of Canonical Coordinates
4.6 Lagrange and Poisson Brackets
4.6.1 The fundamental Lagrange and Poisson brackets
4.6.2 Connection between Lagrange and Poisson brackets
4.7 The Poisson Algebra and its Significance
4.8 Miscellaneous Examples
4.9 Problems without Worked Solutions
5. Symmetries and Transformations
5.1 Introductory Remarks
5.2 Symmetries
5.3 The Galilei Transformation
5.4 Rotation and Rotation Group
5.4.1 Group property of coordinate transformations
5.4.2 The group concept
5.4.3 The orthogonal group O(n)
5.4.4 The groups O(2) and SO(2)
5.4.5 The groups O(3) and SO(3) =: {R}
5.4.6 The unitary groups U(n) and SU(n)
5.4.7 The infinitesimal rotation of a vector
5.5 Rotating Reference Frames
5.6 Definition of Scalars, Vectors, Tensors
5.7 The Theorem of E. Noether
5.8 Canonical Transformations
5.8.1 Generators of canonical transformations
5.8.2 Invariance of Poisson brackets
5.9 Conserved Quantities
5.9.1 Infinitesimal canonical transformations
5.9.2 Infinitesimal transformations and Poisson brackets
5.9.3 Angular momenta and Poisson brackets
5.10 Miscellaneous Examples
5.11 Problems without Worked Solutions
6. Looking Beyond Classical Mechanics
6.1 Introductory Remarks
6.2 Aspects of Classical Statistics
6.2.1 Classical probabilities
6.2.2 The Liouville equation
6.2.3 Probable values of observables
6.3 Spacetime Formulations
6.3.1 Spacetime (Lorentz) transformations
6.3.2 The Poincar´e group
6.3.3 Derivatives
6.4 From Particles to Fields
6.4.1 Euler–Lagrange equations
6.4.2 The Noether theorem
6.4.3 Curved spacetime
6.5 Miscellaneous Examples
6.6 Problems without Worked Solutions
7. Two-Body Central Forces
7.1 Introductory Remarks
7.2 Equations of Motion
7.3 Solution of the Equations
7.4 Differential Equation of the Orbit
7.5 The Kepler Problem
7.6 Tangential Equations of Orbits
7.7 Maxima and Minima of Velocities
7.8 Same Orbit, Different Forces
7.9 Period
7.10 Perihelion Precession of Mercury
7.11 Stability of Circular Orbits
7.12 Scattering in Central Force Fields
7.13 Miscellaneous Examples
7.14 Problems without Worked Solutions
8. Rigid Body Dynamics
8.1 Introductory Remarks
8.2 Moments of Inertia
8.3 Diagonalization and Principal Axes
8.3.1 The ellipsoid of inertia
8.3.2 Transformation to principal axes
8.4 The Equations of Motion
8.5 Miscellaneous Examples I
8.6 Force-free Motion
8.7 The Spinning Top in the Gravitational Field
8.8 Motion Relative to Rotations: Centrifugal and Coriolis Forces
8.9 Miscellaneous Examples II
8.10 Problems without Worked Solutions
9. Small Oscillations and Stability
9.1 Introductory Remarks
9.2 Resonance Frequencies and Normal Modes
9.3 Stability
9.4 Miscellaneous Examples
9.5 Problems without Worked Solutions
10. Motivation of the Theory of Relativity
10.1 Introductory Remarks
10.2 The Weak Equivalence Principle
10.3 Inertial Frames
10.4 The Strong Principle of Equivalence
10.5 The Fundamental Postulate
10.6 Curvature
10.7 Miscellaneous Examples
10.8 Problems without Worked Solutions
11. A Simple Look at Phenomenological Consequences
11.1 Introductory Remarks
11.2 Results of the Special Theory Summarized
11.3 Main Tests of General Relativity
11.3.1 The gravitational redshift
11.3.2 The gravitational deflection of light
11.3.3 The precession of the planet Mercury’s perihelion
12. Aspects of Special Relativity
12.1 Introductory Remarks
12.2 Basics and Physical Motivation of the Lorentz Transformation
12.3 Active and Passive Transformations
12.4 Proper Time and Light Cones
12.5 Lorentz Indices and Transformations
12.5.1 Contravariant vectors and covariant vectors
12.5.2 Tensors
12.6 Lorentz Boosts in Electrodynamics
12.7 Curvature due to Lorentz Contraction
12.8 Covariantization of Newton’s Equation of a Charged Particle
12.9 The Tangent Vector
12.10 Miscellaneous Examples
12.11 Problems without Worked Solutions
13. Equation of Motion of a Particle in a Gravitational Field
13.1 Introductory Remarks
13.2 Equation of Motion
13.3 Reduction to Newton’s Equation
13.4 Rotation Observed from an Inertial Frame
13.5 The Redshift
13.6 Problems without Worked Solutions
14. Tensor Calculus for Riemann Spaces
14.1 Introductory Remarks
14.2 Tensors
14.3 Symmetric and Antisymmetric Tensors
14.4 Definition of Other Important Quantities
14.4.1 Transformation of the metric tensor
14.4.2 Pseudo-tensors and duals
14.4.3 Volume forms
14.5 Covariant Derivatives by the Method of Parallel Transport of a Vector
14.6 Metric Affinity and Christoffel Symbols
14.7 Raising and Lowering of Indices
14.8 Rewriting Co- and Contravariant Derivatives
14.9 Covariant Divergence, Rotation etc.
14.10 Problems without Worked Solutions
15. Einstein’s Equation of the Gravitational Field
15.1 Introductory Remarks
15.2 The Riemann Curvature Tensor
15.3 Bianchi Identities and Ricci–Einstein Tensor
15.4 The Energy–Momentum Tensor
15.4.1 The energy–momentum tensor in electrodynamics
15.4.2 The general case
15.5 Einstein’s Equation of the Gravitational Field
15.6 Newton’s Potential from Einstein’s Equation
15.7 Problems without Worked Solutions
16. The Schwarzschild Solution
16.1 Introductory Remarks
16.2 The Spherical Solution Outside the Source
16.3 The Schwarzschild Solution for Λ = 0
16.4 The Schwarzschild Solution for Λ ≠ 0
16.5 The Relativistic Kepler Problem
16.6 The Light Ray in the Schwarzschild Field
16.7 Problems without Worked Solutions
Appendix A Schwarzschild Orbit Solution
A.1 Introductory Remarks
A.2 The Elliptic Integral
A.3 Evaluating the Elliptic Integral
Appendix B Reissner–Nordstrom Metric
B.1 Introductory Remarks
B.2 The Metric
B.3 The Energy–Momentum Tensor
B.4 The Energy–Momentum Tensor for an Electrostatic Field
B.5 Christoffel Symbols and Riemann Tensor
B.6 The Einstein Equation
B.7 Evaluating the Electrostatic and Gravitational Fields
Bibliography
Index
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