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CONTENTS

Preface................ III

CHAPTER I. THEORY OF VECTORS

I. Operations on vectors

§ 1. Preliminary definitions.................. 1
§ 2. Components of a vector.................. 2
§ 3. Sum and difference of vectors.................. 3
§ 4. Product of a vector by a number.................. 4
§ 5. Components of a sum and product.................. 5
§ 6. Resolution of a vector.................. 6
§ 7. Scalar product.................. 7
§ 8. Vector product.................. 9
§ 9. Product of several vectors.................. 12
§ 10. Vector functions.................. 13
§ 11. Moment of a vector.................. 15

II. Systems of vectors

§ 12. Total moment of a system of vectors.................. 19
§ 13. Parameter.................. 20
§ 14. Equipollent systems.................. 21
§ 15. Vector couple.................. 23
§ 16. Reduction of a system of vectors.................. 23
§ 17. Central axis. Wrench.................. 26
§ 18. Centre of parallel vectors.................. 27
§ 19. Elementary transformations of a system.................. 28

CHAPTER II. KINEMATICS OF A POINT I.

Motion relative to a frame of reference
§ 1. Time.................. 32
§ 2. Frame of reference.................. 32
§ 3. Motion of a point.................. 33
§ 4. Graph of a motion.................. 34
§ 5. Velocity.................. 34
§ 6. Acceleration.................. 36
§ 7. Resolution of the acceleration along a tangent and a normal.................. 39
§ 8. Angular velocity and acceleration.................. 45
§ 9. Plane motion in a polar coordinate system.................. 46
§ 10. Areal velocity.................. 47
§ 11. Dimensions of kinematic magnitudes.................. 49

II. Change of frame of reference

§ 12. Relation among coordinates.................. 52
§ 13. Relation among velocities.................. 56
§ 14. Relations among accelerations.................. 59
§ 15. Determination of relative motion. Motion relative to a point.................. 65

CHAPTER III. DYNAMICS OF A MATERIAL POINT

I. Dynamics of an unconstrained point
§ 1. Basic concepts of dynamics.................. 69
§ 2. Newton's laws of dynamics.................. 71
§ 3. Systems of dynamical units.................. 74
§ 4. Equations of motion.................. 77
§ 5. Motion under the influence of the force of gravity.................. 80
§ 6. Motion in a resisting medium.................. 82
§ 7. Moment of momentum.................. 84
§ 8. Central motion.................. 85
§ 9. Planetary motions.................. 87
§ 10. Work.................. 92
§ 11. Potential force field.................. 96
§ 12. Examples of potential fields.................. 100
§ 13. Kinetic and potential energy.................. 104
§ 14. Motion of a point attracted by a fixed mass.................. 106
§ 15. Harmonic motion.................. 110
§ 16. Conditions for equilibrium in a force field.................. 118

II. Dynamics of a constrained point

§ 17. Equations of motions.................. 121
§ 18. Motion of a constrained point along a curve.................. 123
§ 19. Motion of a constrained point along a surface.................. 127
§ 20. Mathematical pendulum.................. 129
§ 21. Equilibrium of a constrained point.................. 131

III. Dynamics of relative motion

§ 22. Laws of motion.................. 135
§ 23. Examples of motion.................. 136
§ 24. Relative equilibrium.................. 140
§ 25. Motion relative to the earth........... 144

CHAPTER IV. GEOMETRY OF MASSES

I. Systems of points
§ 1. Statical moments.................. 151
§ 2. Centre of mass.................. 152
§ 3. Moments of the second order.................. 157
§ 4. Ellipsoid of inertia. Principal axes of inertia.................. 161
§ 5. Second moments of a plane system.................. 166
II. Solids, surfaces and material lines
§ 6. Density...................... 167
§ 7. Statical moments and moments of inertia. Centre of mass.................. 169
§ 8. Centres of gravity of some curves, surfaces and solids.................. 175
§ 9. Moments of inertia of some curves, surfaces and solids.................. 179

CHAPTER V. SYSTEMS OF MATERIAL POINTS

§ 1. Equations of motion.................. 186
§ 2. Motion of the centre of mass.................. 194
§ 3. Moment of momentum.................. 198
§ 4. Work and potential of a system of points.................. 208
§ 5. Kinetic energy of a system of points.................. 214
§ 6. Problem of two bodies.................. 221
§ 7. Problem of n bodies.................. 224
§ 8. Motion of a body of variable mass.................. 227

CHAPTER VI. STATICS OF A RIGID BODY

I. Unconstrained body
§ 1. Rigid body.................. 231
§ 2. Force.................. 232
§ 3. Hypotheses for the equilibrium of forces.................. 235
§ 4. Transformations of systems of forces.................. 235
§ 5. Conditions for equilibrium of forces.................. 244
§ 6. Graphical statics.................. 249
§ 7. Some applications of the string polygon.................. 253

II. Constrained body

§ 8. Conditions of equilibrium.................. 257
§ 9. Reactions of bodies in contact.................. 258
§ 10. Friction.................. 267
§ 11. Conditions for equilibrium not involving the reaction.................. 270
§ 12. Equilibrium of heavy supported bodies.................. 278
§ 13. Internal forces III. Systems of bodies.................. 284
§ 14. Conditions of equilibrium.................. 286
§ 15. Systems of bars.................. 288
§ 16. Frames.................. 294
§ 17. Equilibrium of heavy cables.................. 302

CHAPTER VII. KINEMATICS OF A RIGID BODY

§ 1. Displacement and rotation of a body about an axis.................. 307
§ 2. Displacements of points of a body in plane motion.................. 310
§ 3. Displacements of the points of a body.................. 312
§ 4. Advancing motion and rotation about an axis.................. 318
§ 5. Distribution of velocities in a rigid body.................. 321
§ 6. Instantaneous plane motion.................. 324
§ 7. Instantaneous space motion.................. 330
§ 8. Rolling and sliding.................. 337
§ 9. Composition of motions of a body.................. 342
§ 10. Analytic representation of the motion of a rigid body.................. 350
§ 11. Resolution of accelerations.................. 357

CHAPTER VIII. DYNAMICS OF A RIGID BODY

§ 1. Work and kinetic energy.................. 360
§ 2. Equations of motion.................. 364
§ 3. Rotation about a fixed axis.................. 374
§ 4. Plane motion.................. 385
§ 5. Angular momentum.................. 393
§ 6. Euler's equations.................. 397
§ 7. Rotation of a body about a point under the action of no forces.................. 399
§ 8. Rotation of a heavy body about a point.................. 406
§ 9. Motion of a sphere on a plane.................. 409
§ 10. Foucault's. gyroscope.................. 412

CHAPTER IX. PRINCIPLE OF VIRTUAL WORK

§ 1. Holonomo-scleronomic systems.................. 418
§ 2. Virtual displacements.................. 422
§ 3. Principle of virtual work.................. 434
§ 4. Determination of the position of equilibrium in a force field.................. 446
§ 5. Lagrange's generalized coordinates.................. 451

CHAPTER X. DYNAMICS OF HOLONOMIC SYSTEMS

§ 1. Holonomic systems.................. 466
§ 2. Non-holonomic systems.................. 467
§ 3. Virtual displacements.................. 468
§ 4. D'Alembert's principle.................. 474
§ 5. Work and kinetic energy in scleronomic systems.................. 478
§ 6. Lagrange's equations of the first kind.................. 480
§ 7. Lagrange's equations of the second kind.................. 483
§ 8. Hamilton's canonical equations.................. 498

CHAPTER XI. VARIATIONAL PRINCIPLES OP MECHANICS

§ 1. Variation without the variation of time.................. 504
§ 2. Hamilton's principle.................. 512
§ 3. Variation with the variation of time.................. 522
§ 4. Maupertuis' principle (of least action)..................... 527

Appendix. Ordinary differential equations of the second order with constant coefficients............. 534

Index......................... 537