Mechanics

Stefan Banach

GDML_Books, (1951), p.

Résumé

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

Détail

EUDML-ID : urn:eudml:doc:219305