 # Mathematical Collection   Send your comments on adress bwm@icm.edu.pl

Monografie Matematyczne

## Volume 24

Mechanics

Stefan Banach

Warszawa-Wrocław 1951

#### Contents

 CHAPTER I THEORY OF VECTORS I. Operations on vectors § 1. Preliminary definitions § 2. Components of a vector § 3. Sum and difference of vectors § 4. Product of a vector by a number § 5. Components of asum and product § 6. Resolution of a vector § 7. Scalar product § 8. Vector product § 9. Product of several vectorsv § 10. Vector functions § 11. Moment of a vector II. Systems of vectors § 12. Total moment of a system of vectors § 13. Parameter § 14. Equipollent systems § 15. Vector couple § 16. Reduction of a system of vectors § 17. Central axis § 18. Centre of parallel vectors § 19. Elementary transformations of a system CHAPTER II KINEMATICS OF A POINT I. Motion relative to a frame of reference § 1. Time § 2. Frame of reference § 3. Motion of a point § 4.Graph of a motion § 5. Velocity § 6. Acceleration § 7. Resolution of the acceleration along a tangent and a normal § 8. Angular velocity and acceleration § 9. Plane motion in a polar coordinate system § 10. Areal velocity § 11. Dimensions of kinematic magnitudes II. Change of frame of reference § 12. Relation among coordinates § 13. Relation among velocities § 14. Relations among accelerations § 15. Determination of relative motion CHAPTER III DYNAMICS OF A MATERIAL POINT I. Dynamics of an unconstrained point § 1. Basic concepts of dynamics § 2. Newton's laws of dynamics § 3. Systems of dynamical units § 4. Equations of motion § 5. Motion under the influence of the force of gravity § 6. Motion in a resisting medium § 7. Moment of momentum § 8. Central motion § 9. Planetary motions § 10.Work § 11. Potential force field § 12. Examples of potential fields § 13. Kinetic and potential energy § 14. Motion of a point attracted by af ixed mass § 15. Harmonic motion § 16. Conditions for equilibrium in aforcefield II. Dynamics of a constrained point § 17. Equations of motions § 18. Motion of a constrained point along a curvece § 19. Motion of a constrained point along a surface § 20. Mathematical pendulum § 21. Equilibrium of a constrained point III. Dynamics of relative motion § 22. Laws of motion § 23. Examples of motion § 24. Relative equilibrium § 25. Motion relative to the earth CHAPTER IV GEOMETRY OF MASSES I. Systems of points § 1.Statical moments § 2. Centre of mass § 3. Moments of the second order § 4. Ellipsoid of inertia § 5. Second moments of a plane system II. Solids, surfaces and material lines § 6. Density § 7. Statical moments and moments of inertia § 8. Centres of gravity of some curves, surfaces and solids § 9. Moments of inertia of some curves, surfaces and solids CHAPTER V SYSTEMS OF MATERIAL POINTS § 1. Equations of motion § 2. Motion of the centre of mass § 3. Moment of momentum § 4. Work and potential of a system of points § 5. Kinetic energy of a system of points § 6. Problem of two bodies § 7.Problem of n bodies § 8. Motion of a body of variable mass CHAPTER VI STATICS OF A RIGID BODY I. Unconstrained body § 1. Rigid body § 2. Force § 3. Hypotheses for the equilibrium offorces § 4.Transformations of systems of forces § 5. Conditions for equilibrium of forces § 6. Graphical statics § 7. Some applications of the string polygon II. Constrained body § 8. Conditions of equilibrium § 9. Reactions of bodies in contact § 10. Friction § 11. Conditions for equilibrium not involving the reaction § 12. Equilibrium of heavy supported bodies § 13. Internal forces III. Systems of bodies § 14. Conditions of equilibrium § 15. Systems of bars § 16. Frames § 17. Equilibrium of heavy cables CHAPTER VII KINEMATICS OF A RIGID BODY § 1. Displacement and rotation of a body about an axis § 2. Displacements of points of a body in plane motion § 3. Displacements of the points of a body § 4. Advancing motion and rotation about an axis § 5. Distribution of velocities in a rigid body § 6. Instantaneous plane motion § 7.Instantaneous space motion § 8. Rolling and sliding § 9. Composition of motions of a body § 10. Analytic representation of the motion of a rigid body § 11. Resolution of accelerations CHAPTER VIII DYNAMICS OF A RIGID BODY § 1. Work and kinetic energy § 2. Equations of motion § 3. Rotation about a fixed axis § 4. Plane motion § 5.Angular momentum § 6. Euler's equations § 7. Rotation of a body about a point under the action of no forces § 8. Rotation of a heavy body about a point § 9. Motion of a sphere on a plane § 10. Foucault's. gyroscope CHAPTER IX PRINCIPLE OF VIRTUAL WORK § 1. Holonomo-scleronomic systems § 2.Virtual displacements § 3. Principle of virtual work § 4. Determination of the position of equilibrium in a force field § 5. Lagrange's generalized coordinates CHAPTER X DYNAMICS OF HOLONOMIC SYSTEMS § 1. Holonomic systems § 2. Non-holonomic systems § 3. Virtual displacements § 4. D'Alembert's principle § 5. Work and kinetic energy in scleronpmic systems § 6. Lagrange's equations of the first kind § 7. Lagrange's equations of the second kind § 8. Hamilton's canonical equations CHAPTER XI VARIATIONAL PRINCIPLES OP MECHANICS § 1. Variation without the variation of time § 2. Hamilton's principle § 3. Variation with the variation of time § 4. Maupertuis' principle (of least action) Materiały redakcyjne Preface, appendix, index, contents