Mechanics

Banach, Stefan

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa-Wrocław 1951), 1951

Abstract

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CONTENTSPreface................ IIICHAPTER I. THEORY OF VECTORSI. 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.................. 15II. 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.................. 28CHAPTER 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.................. 49II. 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.................. 65CHAPTER III. DYNAMICS OF A MATERIAL POINTI. 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.................. 118II. 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.................. 131III. Dynamics of relative motion§ 22. Laws of motion.................. 135§ 23. Examples of motion.................. 136§ 24. Relative equilibrium.................. 140§ 25. Motion relative to the earth........... 144CHAPTER IV. GEOMETRY OF MASSESI. 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.................. 166II. 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.................. 179CHAPTER 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.................. 227CHAPTER VI. STATICS OF A RIGID BODYI. 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.................. 253II. 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.................. 302CHAPTER 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.................. 357CHAPTER 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.................. 412CHAPTER 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.................. 451CHAPTER 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.................. 498CHAPTER 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)..................... 527Appendix. Ordinary differential equations of the second order with constant coefficients............. 534Index......................... 537

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Banach, Stefan. Mechanics. Warszawa-Wrocław 1951: Instytut Matematyczny Polskiej Akademi Nauk, 1951. <http://eudml.org/doc/219305>.

@book{Banach1951,
abstract = {CONTENTSPreface................ IIICHAPTER I. THEORY OF VECTORSI. 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.................. 15II. 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.................. 28CHAPTER 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.................. 49II. 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.................. 65CHAPTER III. DYNAMICS OF A MATERIAL POINTI. 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.................. 118II. 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.................. 131III. Dynamics of relative motion§ 22. Laws of motion.................. 135§ 23. Examples of motion.................. 136§ 24. Relative equilibrium.................. 140§ 25. Motion relative to the earth........... 144CHAPTER IV. GEOMETRY OF MASSESI. 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.................. 166II. 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.................. 179CHAPTER 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.................. 227CHAPTER VI. STATICS OF A RIGID BODYI. 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.................. 253II. 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.................. 302CHAPTER 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.................. 357CHAPTER 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.................. 412CHAPTER 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.................. 451CHAPTER 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.................. 498CHAPTER 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)..................... 527Appendix. Ordinary differential equations of the second order with constant coefficients............. 534Index......................... 537},
author = {Banach, Stefan},
keywords = {Mechanics},
language = {eng},
location = {Warszawa-Wrocław 1951},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Mechanics},
url = {http://eudml.org/doc/219305},
year = {1951},
}

TY - BOOK
AU - Banach, Stefan
TI - Mechanics
PY - 1951
CY - Warszawa-Wrocław 1951
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSPreface................ IIICHAPTER I. THEORY OF VECTORSI. 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.................. 15II. 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.................. 28CHAPTER 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.................. 49II. 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.................. 65CHAPTER III. DYNAMICS OF A MATERIAL POINTI. 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.................. 118II. 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.................. 131III. Dynamics of relative motion§ 22. Laws of motion.................. 135§ 23. Examples of motion.................. 136§ 24. Relative equilibrium.................. 140§ 25. Motion relative to the earth........... 144CHAPTER IV. GEOMETRY OF MASSESI. 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.................. 166II. 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.................. 179CHAPTER 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.................. 227CHAPTER VI. STATICS OF A RIGID BODYI. 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.................. 253II. 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.................. 302CHAPTER 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.................. 357CHAPTER 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.................. 412CHAPTER 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.................. 451CHAPTER 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.................. 498CHAPTER 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)..................... 527Appendix. Ordinary differential equations of the second order with constant coefficients............. 534Index......................... 537
LA - eng
KW - Mechanics
UR - http://eudml.org/doc/219305
ER -

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