Lectures on Theoretical Mechanics Dietrich Belitz Department of Physics University of Oregon Contents Acknowledgments, and Disclaimer 0 1 Mathematical principles of mechanics 1 x 1 Philosophical comments . 1 x 1.1 The role of theoretical physics . 1 x 1.2 The structure of a physical theory . 2 x 2 Review of basic mathematical concepts and notation . 2 x 2.1 Number sets and functions . 2 x 2.2 Differentiation . 3 x 2.3 Stationary points . 5 x 2.4 Ordinary differential equations . 7 x 2.5 Integration . 8 x 2.6 Paths and path functionals . 9 x 2.7 Surfaces . 12 x 2.8 Line and surface integrals . 12 x 2.9 The implicit function theorem . 15 x 3 Calculus of variations . 16 x 3.1 Three classic problems . 16 x 3.2 Path neighborhoods . 17 x 3.3 A fundamental lemma . 18 x 3.4 The Euler-Lagrange equations . 19 x 4 General discussion of the Euler-Lagrange equations . 21 x 4.1 First integrals of the Euler-Lagrange equations . 21 x 4.2 Example: The brachistochrone problem . 22 x 4.3 Covariance of the Euler-Lagrange equations . 25 x 4.4 Gauge invariance . 26 x 4.5 Nöther’s theorem . 27 x 5 The principle of least action . 30 x 5.1 The axioms of classical mechanics . 30 x 5.2 Conservation laws . 31 x 6 Problems for Chapter 1 . 32 2 Mechanics of point masses 39 x 1 Point masses in potentials . 39 x 1.1 A single free particle . 39 i ii Contents x 1.2 Galileo’s principle of relativity . 41 x 1.3 Potentials . 42 x 1.4 The equations of motion . 44 x 1.5 Systems of point masses . 46 x 2 Simple examples for the motion of point masses . 49 x 2.1 Galileo’s law of falling bodies . 49 x 2.2 Particle on an inclined plane . 53 x 2.3 Particle on a rotating pole . 54 x 2.4 The harmonic oscillator . 55 x 3 One-dimensional conservative systems . 57 x 3.1 Definition of the model . 57 x 3.2 Solution to the equations of motion . 57 x 3.3 Unbounded motion . 58 x 3.4 Bounded motion . 62 x 3.5 Equilibrium positions . 63 x 4 Motion in a central field . 64 x 4.1 Reduction to a one-dimensional problem . 64 x 4.2 Kepler’s second law . 65 x 4.3 Radial motion . 67 x 4.4 Equation of the orbit . 68 x 4.5 Classification of orbits . 69 x 4.6 Kepler’s problem . 72 x 4.7 Kepler’s laws of planetary motion . 78 x 5 Introduction to perturbation theory . 79 x 5.1 General concept . 79 x 5.2 Digression - Fourier series . 80 x 5.3 Oscillations and Fourier series . 83 x 5.4 Summary thus far, and the anharmonic oscillator . 84 x 5.5 Perturbation theory for the central field problem . 92 x 5.6 Perturbations of Kepler’s problem . 94 x 5.7 Digression - introduction to potential theory . 96 x 5.8 Mercury’s perihelion advance due to an oblate sun . 102 x 6 Scattering theory . 104 x 6.1 Scattering experiments . 104 x 6.2 Classical theory for the scattering cross section . 106 x 6.3 Scattering by a central potential . 107 x 6.4 Rutherford scattering . 108 x 6.5 Scattering by a hard sphere . 110 x 7 N-particle systems . 111 x 7.1 Closed N-particle systems . 111 x 7.2 The two-body problem . 116 x 8 Problems for Chapter 2 . 119 Contents iii 3 The rigid body 128 x 1 Mathematical preliminaries - vector spaces and tensor spaces . 128 x 1.1 Metric vector spaces . 128 x 1.2 Coordinate transformations . 131 x 1.3 Proper coordinate systems . 132 x 1.4 Proper coordinate transformations . 133 x 1.5 Tensor spaces . 135 x 2 Orthogonal transformations . 138 x 2.1 Three-dimensional Euclidean space . 138 x 2.2 Special orthogonal transformations - the group SO (3) . 140 x 2.3 Rotations about a fixed axis . 142 x 2.4 Infinitesimal rotations - the generators of SO (3) . 144 x 2.5 Euler angles . 145 x 3 The rigid body model . 148 x 3.1 parameterization of a rigid body . 148 x 3.2 Angular velocity . 150 x 3.3 The inertia tensor . 155 x 3.4 Classification of rigid bodies . 157 x 3.5 Angular momentum of a rigid body . 157 x 3.6 The continuum model of rigid bodies . 159 x 3.7 The Euler equations . 160 x 4 Simple examples of rigid-body motion . 161 x 4.1 The physical pendulum . 161 x 4.2 Cylinder on an inclined plane . 162 x 4.3 The force-free symmetric top . 164 x 4.4 The force-free asymmetric top . 165 x 5 Problems for Chapter 3 . 170 Acknowledgments These lectures are based in part on notes taken in lectures delivered by Prof. Wolfgang Götze at the Technical University Munich between 1976 and 1985. In their current form they have been taught as a 15-week course for first-year graduate students at the University of Oregon. The original handwritten notes were typeset by Spencer Alexander during the academic year 2013/14. Julian Smith helped with the first round of proofreading. Thanks are due to everyone who has alerted me to typos since. All remaining mistakes and misconceptions are, of course, the author’s responsibility. Disclaimer These notes are still a work in progress. If you notice any mistakes, whether it’s trivial.