Classical Mechanics And Relativity

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