# Mechanics

### Lectures on Theoretical Physics

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Mechanics: Lectures on Theoretical Physics, Volume I covers a general course on theoretical physics. The book discusses the mechanics of a particle; the mechanics of systems; the principle of virtual work; and d’alembert’s principle. The text also describes oscillation problems; the kinematics, statics, and dynamics of a rigid body; the theory of relative motion; and the integral variational principles of mechanics. Lagrange’s equations for generalized coordinates and the theory of Hamilton are also considered. Physicists, mathematicians, and students taking Physics courses will find the book invaluable.

## Table of Contents

Foreword to Sommerfelds's Course

Preface to the First Edition

Introduction

Chapter I. Mechanics of a Particle

1. Newton's Axioms

2. Space, Time and Reference Systems

3. Rectilinear Motion of a Mass Point

Examples:

(1) Free Fall Near Earth's Surface (Falling Stone)

(2) Free Fall From a Great Distance (Meteor)

(3) Free Fall in Air

(4) Harmonic Oscillations

(5) Collision of Two Particles

4. Variable Masses

5. Kinematics and Statics of a Single Mass Point in a Plane and in Space

(1) Plane Kinematics

(2) The Concept of Moment in Plane Statics and Kinematics

(3) Kinematics in Space

(4) Statics in Space; Moment of Force About a Point and About an Axis

6. Dynamics (Kinetics) of the Freely Moving Mass Point; Kepler Problem; Concept of Potential Energy

(1) Kepler Problem with Fixed Sun

(2) Kepler Problem Including Motion of the Sun

(3) When Does a Force Field Have a Potential?

Chapter II. Mechanics of Systems, Principle Of Virtual Work, and d'alembert's Principle

7. Degrees of Freedom and Virtual Displacements of a Mechanical System; Holonomic and Non-holonomic Constraints

8. The Principle of Virtual Work

9. Illustrations of the Principle of Virtual Work

(1) The Lever

(2) Inverse of the Lever: Cyclist, Bridge

(3) The Block and Tackle

(4) The Drive Mechanism of a Piston Engine

(5) Moment of a Force About an Axis and Work in a Virtual Rotation

10. D'Alembert's Principle; Introduction of Inertial Forces

11. Application of d'Alembert's Principle to the Simplest Problems

(1) Rotation of a Rigid Body About a Fixed Axis

(2) Coupling of Rotational and Translational Motion

(3) Sphere Rolling on Inclined Plane

(4) Mass Guided Along Prescribed Trajectory

12. Lagrange's Equations of the First Kind

13. Equations of Momentum and of Angular Momentum

(1) Equation of Momentum

(2) Equation of Angular Momentum

(3) Proof Using the Coordinate Method

(4) Examples

(5) Mass Balancing of Marine Engines

(6) General Rule on the Number of Integrations Feasible in a Closed System

14. The Laws of Friction

(1) Static Friction

(2) Sliding Friction

Chapter III. Oscillation Problems

15. The Simple Pendulum

16. The Compound Pendulum

Supplement: A Rule Concerning Moments of Inertia

17. The Cycloidal Pendulum

18. The Spherical Pendulum

19. Various Types of Oscillations. Free and Forced, Damp and Undamped Oscillations

20. Sympathetic Oscillations

21. The Double Pendulum

Chapter IV. The Rigid Body

22. Kinematics of Rigid Bodies

23. Statics of Rigid Bodies

(1) The Conditions of Equilibrium

(2) Equipollence ; the Reduction of Force Systems

(3) Change of Reference Point

(4) Comparison of Kinematics and Statics

Supplement: Wrenches and Screw Displacements

24. Linear and Angular Momentum of a Rigid Body. Their Connection with Linear and Angular Velocity

25. Dynamics of a Rigid Body. Survey of its Forms of Motion

(1) The Spherical Top Under No Forces

(2) The Symmetrical Top Under No Forces

(3) The Unsymmetrical Top Under No Forces

(4) The Heavy Symmetrical Top

(5) The Heavy Unsymmetrical Top

26. Euler's Equations. Quantitative Treatment of the Top Under No Forces

(1) Euler's Equations of Motion

(2) Regular Precession of the Symmetrical Top Under No Forces andEuler's Theory of Polar Fluctuations

(3) Motion of an Unsymmetrical Top Under No Forces. Examination of its Permanent Rotations as to Stability

27. Demonstration Experiments Illustrating the Theory of the Spinning Top; Practical Applications

(1) The Gyrostabilizer and Related Topics

(2) The Gyrocompass

(3) Gyroscopic Effects in Railroad Wheels and Bicycles

Supplement: The Mechanics of Billiards

(a) High and Low Shots, 158—(b) Follow Shots and Draw Shots,

(c) Trajectories with "English" Under Horizontal Impact,

(d) Parabolic Path Due to Shot with Vertical Component,

Chapter V. Relative Motion

28. Derivation of the Coriolis Force in a Special Case

29. The General Differential Equations of Relative Motion

30. Free Fall on the Rotating Earth; Nature of the Gyroscopic Terms

31. Foucault's Pendulum

32. Lagrange's Case of the Three-Body Problem

Chapter VI. Integral Variational Principles Of Mechanics and Lagrange's Equations For Generalized Coordinates

33. Hamilton's Principle

34. Lagrange's Equations for Generalized Coordinates

35. Examples Illustrating the Use of Lagrange's Equations

(1) The Cycloidal Pendulum

(2) The Spherical Pendulum

(3) The Double Pendulum

(4) The Heavy Symmetrical Top

36. An Alternate Derivation of Lagrange's Equations

37. The Principle of Least Action

Chapter VII. Differential Variational Principles Of Mechanics

38. Gauss' Principle of Least Constraint

39. Hertz's Principle of Least Curvature

40. A Digression on Geodesies

Chapter VIII. The Theory Of Hamilton

41. Hamilton's Equations

(1) Derivation of Hamilton's Equations from Lagrange's Equations

(2) Derivation of Hamilton's Equations from Hamilton's Principle

42. Routh's Equations and Cyclic Systems

43. The Differential Equations for Non-Holonomic Velocity Parameters

44. The Hamilton-Jacobi Equation

(1) Conservative Systems

(2) Dissipative Systems

45. Jacobi's Rule on the Integration of the Hamilton-Jacobi Equation

46. Classical and Quantum-Theoretical Treatment of the Kepler Problem

Problems

For Chapter I

I.1, I.2, I.3. Elastic collision

I.4. Inelastic collision between an electron and an atom

I.5. Rocket to the moon

I.6. Water drop falling through saturated atmosphere

I.7. Falling chain

I.8. Falling rope

I.9. Acceleration of moon due to earth's attraction

I.10. The torque as vector quantity

I.11. The hodograph of planetary motion

I.12. Parallel beam of electrons passing through the field of an ion. Envelope of the trajectories

I.13. Elliptical trajectory under the influence of a central force directly proportional to the distance

I.14. Nuclear disintegration of lithium

I.15. Central collisions between neutrons and atomic nuclei; effect of a block of paraffin

I.16. Kepler's equation

For Chapter II

II.1. Non-holonomic conditions of a rolling wheel

II.2. Approximate design of a flywheel for a double-acting one-cylinder steam engine

II.3. Centrifugal force under increased rotation of the earth

II.4. Poggendorff's experiment

II.5. Accelerated inclined plane

II.6. Products of inertia for the uniform rotation of an unsymmetrical body about an axis

II.7. Theory of the Yo-yo

II.8. Particle moving on the surface of a sphere

For Chapter III

III.1. Spherical pendulum under infinitesimal oscillations

III.2. Position of the resonance peak of forced damped oscillations

III.3. The galvanometer

III.4. Pendulum under forced motion of its point of suspension

III.5. Practical arrangement of coupled pendulums

III.6. The oscillation quencher

For Chapter IV

IV.1. Moments of inertia of a plane mass distribution

IV.2. Rotation of the top about its principal axes

IV.3. High and low shots in a billiard game. Follow shot and draw shot

IV.4. Parabolic motion of a billiard ball

For Chapter V

V.1. Relative motion in a plane

V.2. Motion of a particle on a rotating straight line

V.3. The sleigh as the simplest example of a non-holonomic system

For Chapter VI

VI.1. Example illustrating Hamilton's principle

VI.2. Relative motion in a plane and motion on a rotating straight line

VI.3. Free fall on the rotating earth and Foucault's pendulum

VI.4. "Wobbling" of a cylinder rolling on a plane support

VI.5. Differential of an automobile

Hints for Solving the Problems

Index

## Product details

- No. of pages: 304
- Language: English
- Copyright: © Academic Press 1952
- Published: January 1, 1952
- Imprint: Academic Press
- eBook ISBN: 9781483220284

## About the Author

### Arnold Sommerfeld

#### Affiliations and Expertise

University of Munich