Home Page of Stefan Banach

Home Page of Stefan Banach

"Mechanics"
by Stefan Banach
"Monografie Matematyczne" series, vol. 25, Polish Mathematical Society, Warszawa-Wrocław, 1951, p. IV+546.
CONTENTS [PDF] 
Preface [PDF] 
My lectures in mechanics, given for many years at the Jan Kazimierz University and the Lwów Polytechnic Institute, consisted of the contents of this work.
I have limited myself to the mechanics of a system of material points and a rigid body. The material covered is suitable in general for university requirements, however, keeping in mind the needs of the students at the Polytechnic Institute, chapter VI which treats of the statics of a rigid body was worked out so that it could be accessible without a knowledge of kinematics and dynamics. It can be read immediately after chapter I when supplemented with several facts according to instructions included in the footnote on p. 231. Taking into account the requirements in mechanics at the Polytechnic Institute, I have also given in chapter VI certain information from engineering mechanics.
The mathematics necessary to understand this work in its entirely is limited to the elements of analytic geometry as well as to the differential and integral calculus. Other necessary notions and theorems have been given in the text in order not to send the reader to works too specialized. In particular, I give in the Appendix (at the end of the work) a method of solving ordinary differential equations of second order with constant coefficients which arise frequently in mechanics.
I have endeavoured to give in this book as easy an exposition as possible. The more difficult considerations are illustrated by many examples. Problems to be solved have not been inserted in the text; the reader will find them in most textbooks and in the collections of problems which are given below. I have also considered it unnecessary to burden the contents with the names of the authors of particular theorems or examples, since I consider the material contained in this book as classical. The reader will find detailed bibliographic instructions in corresponding articles of volume IV of the Enzyklopädie der mathematischen Wissenschaften (Teubner, Leipzig 1901-1935) as well as in volume V of the work Handbuch der Physik (Berlin 1927), and historical data and critical remarks in the work of E. Mach Die Mechanik in ihrer Entwicklung (9 Aufl., Leipzig 1933). Here I limit myself to the presentation from literature of a number of the more important works, namely: (...)
It is a pleasure for me to thank Dr Bronisław Knaster for his effective help in connection with the publishing of this book, Dr Edward Otto for making the drawings and Dr Antoni Zygmund for help in the corrections.
Stefan Banach.
Lwów, January, 1938.
CHAPTER I. THEORY OF VECTORS [PDF] 
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. Distributive law. Associative law. Square of a vector. Analytic representation of a scalar product ... 7
§ 8. Vector product. Change of order of factors. Associative law. Distributive law with respect to a sum. Components of a vector product ... 9
§ 9. Product of several vectors ... 12
§ 10. Vector functions. Limit. Continuity. Derivative. Vector functions of many variables ... 13
§ 11. Moment of a vector. Moment of a vector with respect to a point. Moment as a vector product. Moment of a sum of vectors having a common origin. Components of a moment. Moment of a vector with respect to a line ... 15
II. Systems of vectors.
§ 12. Total moment of a system of vectors ... 19
§ 13. Parameter ... 20
§ 14. Equipollent systems. Systems equipollent to zero. System of three vectors equipollent to zero ... 21
§ 15. Vector couple ... 23
§ 16. Reduction of a system of vectors. Reduction theorem. Table ... 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 [PDF] 
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. Velocity as a derivative of the path ... 34
§ 6. Acceleration. Uniform straight line motion. Hodograph ... 36
§ 7. Resolution of the acceleration along a tangent and a normal. Motion along a plane curve. Motion along a space curve. Uniform motion. Uniformly accelerated motion. Motion along a cycloid ... 39
§ 8. Angular velocity and acceleration. Angular velocity vector ... 45
§ 9. Plane motion in a polar coordinate system ... 46
§ 10. Areal velocity ... 47
§ 11. Dimensions of kinematic magnitudes. General definition of dimension. Determination of dimension ... 49
II. Change of frame of reference.
§ 12. Relation among coordinates.Motion along a holix. ... 52
§ 13. Relation among velocities ... 56
§ 14. Relations among accelerations. Acceleration of Coriolis ... 59
§ 15. Determination of relative motion. Motion relative to a point. ... 65
CHAPTER III. DYNAMICS OF A MATERIAL POINT [PDF] 
I. Dynamics of an unconstrained point.
§ 1. Basic concepts of dynamics. Inertial frame, absolute time. Mass and force. Material point. ... 69
§ 2. Newton's laws of dynamics. Laws of motion. Equilibrium of a point and forces. Force of inertia. D'Alembert's principle ... 71
§ 3. Systems of dynamical units. C. g. s. system. Measurement of masses and forces. Metric gravitational system of units. Dimensions of dynamical magnitudes ... 74
§ 4. Equations of motion ... 77
§ 5. Motion under the influence of the force of gravity. Vertical projection. Oblique projection ... 80
§ 6. Motion in a resisting medium. Vortical projection. Oblique projection ... 82
§ 7. Moment of momentum. Principle of conservation (if areas ... 84
§ 8. Central motion. Binet's formula ... 85
§ 9. Planetary motions. Kepler's laws. Corollaries of Kepler's laws. Law of universal gravitation. Mass of the earth. Kepler's equation... 87
§ 10. Work. Constant force. Variable force. Work of a sum of forces. Dimension and units of work ... 92
§ 11. Potential force field. Stress field. Lines of forces. Definition of a potential field. Potential. Dimension of the potential. Relation between force and potential. Potential surfaces ... 96
§ 12. Examples of potential fields. Constant field. Central field. Newtonian gravitational field. Axial field. Sum of potential fields ... 100
§ 13. Kinetic and potential energy ... 104
§ 14. Motion of a point attracted by a fixed mass. Motion along a conic. Motion along a straight line ... 106
§ 15. Harmonic motion. Simple harmonic motion. Table of position, velocity and acceleration of the point. Plane harmonic motion. Damped harmonic, motion. Forced harmonic motion. Lissajous' curves ... 110
§ 16. Conditions for equilibrium in a force field. Stable equilibrium. Dirichlet's theorem ... 118
II. Dynamics of a constrained point.
§ 17. Equations of motions. Reaction. Equations of motion. Kinetic energy ... 121
§ 18. Motion of a constrained point along a curve. Motion along a plane curve. Motion along a space curve. Motion of a heavy constrained point ... 23
§ 19. Motion of a constrained point along a surface ... 127
§ 20. Mathematical pendulum ... 129
§ 21. Equilibrium of a constrained point. Stable equilibrium. Equilibrium in a potential field ... 131
III. Dynamics of relative motion.
§ 22. Laws of motion ... 135
§ 23. Examples of motion. Advancing motion of a frame. Rotary motion of a frame ... 136
§ 24. Relative equilibrium. Relative equilibrium in a frame moving with an advancing motion ... 140
§ 25. Motion relative to the earth. Force of gravity. Magnitude and direction of the earth's force of attraction. Force of Coriolis. Deviation to the east of a falling body. Foucault's pendulum ... 144
CHAPTER IV. GEOMETRY OF MASSES [PDF] 
I. Systems of points.
§ 1. Statical moments. Statical moment of a point. Statical moment of a system of points ... 161
§ 2. Centre of mass. Centre of mass of two systems of points. Plane system of points. Linear system of points. Centre of mass of two points. Symmetric systems of points ... 152
§ 3. Moments of the second order. Moment of inertia. Product of inertia. Radius of gyration. Concentrated mass. Moments of inertia with respect to parallel lines. Products of inertia with respect to parallel planes ... 157
§ 4. Ellipsoid of inertia. Principal axes of inertia. Determination of principal axes of inertia ... 161
§ 5. Second moments of a plane system ... 166
II. Solids, surfaces and material lines.
§ 6. Density. Calculation of mass. Material surface, material line ... 167
§ 7. Statical moments and moments of inertia. Centre of mass. Geometric solids, surfaces and curves. Guldin's rules. Moments of inertia and products of inertia... 169
§ 8. Centres of gravity of some curves, surfaces and solids. Broken line. Triangle. Trapezoid. Polygon. Sector of a circle. Segment of a circle. Prism. Cylinder. Pyramid. Cone ... 175
§ 9. Moments of inertia of some curves, surfaces and solids. Segment. Rectangle. Square. Trapezoid. Triangle. Parallelogram. Rectangular parallelepiped. Circumference of a circle. Circle. Surface of a sphere. Sphere. Cylinder of a revolution. Cone of revolution ... 179
CHAPTER V. SYSTEMS OF MATERIAL POINTS [PDF] 
§ 1. Equations of motion. Unconstrained systems. Internal and external forces. Equilibrium of a system of points. D'Alembert's principle. Constrained systems. Rigid system. Atwood's machine ... 186
§ 2. Motion of the centre of mass. Kinematic properties of the centre of mass. Resultant of a system of weights. Dynamic properties of the centre of mass . . 194
§ 3. Moment of momentum. Angular momentum with respect to a point. Angular momentum in an advancing motion. Angular momentum in a motion relative to the centre of mass. Angular momentum with respect to an axis. Dynamic properties of angular momentum. Motion in a gravitational field. Rotation of a system about an axis. Angular momentum in relative motion ... 198
§ 4. Work and potential of a system of points. Work. Work equal to zero. Potential of a system. Potential of the force of gravity. Potential of the internal forces ... 208
§ 5. Kinetic energy of a system of points. Kinetic energy of a system in an advancing motion. Kinetic energy in a rotating motion about an axis. Theorem of König. Principle of the equivalence of work and kinetic energy. Kinetic energy in relative motion ... 214
§ 6. Problem of two bodies ... 221
§ 7. Problem of n bodies. Problem of three bodies ... 224
§ 8. Motion of a body of variable mass... 227
CHAPTER VI. STATICS OF A RIGID BODY [PDF] 
I. Unconstrained body.
§ 1. Rigid body. Rigid systems of material points ... 231
§ 2. Force. Point of application of a force. Moment of o force with respect to a point. Moment of a force with respect to an axis. Equilibrium of forces ... 232
§ 3. Hypotheses for the equilibrium of forces ... 235
§ 4. Transformations of systems of forces. Change of the point of application of a force. Law of composition and resolution of forces. Equipollent systems. Force couple. Reduction of a system of forces. Plane system of forces. System of parallel forces. Gravitational forces. Systems of couples. . . . 235
§ 5. Conditions for equilibrium of forces. Analytic form of the conditions for equilibrium. Plane systems of forces ... 244
§ 6. Graphical statics. String polygon. Composition of forces. String polygon. Resultant of a part of a system ... 249
§ 7. Some applications of the string polygon. Determination of the reactions at the points of support. Determination of the moment of forces. Determination of the centre of gravity and the statical moment of plane figures. ... 253
II. Constrained body.
§ 8. Conditions of equilibrium. ... 207
§ 9. Reactions of bodies in contact. Normal and tangential reactions. Supports. Centre of pressure. Reactions of a string... 258
§ 10. Friction. ... 267
§ II. Conditions for equilibrium not involving the reaction. Body with one fixed point. Body with a fixed axis. Plane motion of a body. Lever. Determination of the reactions acting on a fixed axis ... 270
§ 12. Equilibrium of heavy supported bodies. Body supported at two points. Body supported at n > 2 points ... 278
§ 13. Internal forces. ... 284
III. Systems of bodies.
§ 14. Conditions of equilibrium ... 286
§ 15. Systems of bars. Stresses in bars. Pin-connections. Systems of bars. Decimal balance ... 288
§ 16. Frames. Plane frame. Analytical method of determining stresses in a frame. Determination of stresses in a frame (by means of force diagrams). Determination of stresses by means of sections ... 294
§ 17. Equilibrium of heavy cables. Chain. Cable. Loaded cable. ... 302
CHAPTER VII. KINEMATICS OF A RIGID BODY [PDF] 
§ 1. Displacement and rotation of a body about an axis. Parallel displacement or translation. Rotation about an axis ... 307
§ 2. Displacements of points of a body in plane motion. Rotation about a point. I Theorem of Euler. Plane motion of a body... 310
§ 3. Displacements of the points of a body. II Theorem of Euler. Chasle's theorem. Twist ... 312
§ 4. Advancing motion and rotation about an axis. Advancing motion. Rotation about .an axis' ... 318
§ 5. Distribution of velocities in a rigid body. Relations among the velocities of the points of a body. Velocities of points of a straight line and a plane. Instantaneous motion of a rigid body ... 321
§ 6. Instantaneous plane motion. Determination of the instantaneous centre of rotation ... 324
§ 7. Instantaneous space motion. Rotation about a point. Instantaneous motion in the general case. Velocity of transport. Instantaneous twist. Determination of the motion of a body ... 330
§ 8. Rolling and sliding. Curve of instantaneous centres. Cone of instantaneous axes. Surface of central axes ... 337
§ 9. Composition of motions of a body. Two simultaneous rotations. Composition of several simultaneous rotations. Relative motion of a body. Steady precession ... 342
§ 10. Analytic representation of the motion of a rigid body. Instantaneous angular velocity. Central axis. Plane motion. Euler's angles. Euler's angles in a steady precession ... 350
§ 11. Resolution of accelerations. Plane motion. Motion in space ... 357
CHAPTER VIII. DYNAMICS OF A RIGID BODY [PDF] 
§ 1. Work and kinetic energy. Dynamical magnitudes. Work. Kinetic energy . 360
§ 2. Equations of motion. Motion of the centre of mass. Principle of angular momentum. Principle of kinetic energy. D'Alembert's principle. Advancing motion of a body. Conditions of equilibrium. Reactions of bodies in contact. Work of the friction ... 364
§ 3. Rotation about a fixed axis. Atwood's machine. Compound pendulum. Determination of the reaction on an axis of rotation. Axis of rotation as a central axis of inertia. Centre of percussion ... 374
§ 4. Plane motion. Plane motion of a plane figure. Plane motion of a body ... 385
§ 5. Angular momentum. Angular momentum with respect to the centre of mass of a body or with respect to its fixed point. Derivative of the angular momentum ... 393
§ 6. Euler's equations. Motion of an unconstrained rigid body. ... 397
§ 7. Rotation of a body about a point under the action of no forces. Angular momentum and kinetic energy. Rotation about a spherical point. Rotation about a point whose ellipsoid of inertia is an ellipsoid of revolution.
Determination of Euler's angles. Rotation of a body about a point in the general case... 399
§ 8. Rotation of a heavy body about a point ... 406
§ 9. Motion of a sphere on a plane ... 409
§ 10. Foucault's gyroscope. Motion of the axis of symmetry in a meridional plane. Motion of the axis in a horizontal plane ... 412
CHAPTER IX. PRINCIPLE OF VIRTUAL WORK [PDF] 
§ I. Holonomo-scleronomic systems. Bilateral constraints. Unilateral constraints. Degrees of freedom of a system ... 418
§ 2. Virtual displacements. Point on a surface. Point on a curve. Holonomo-scleronomic systems. Bilateral constraints. Unconstrained rigid body. Fixed point. Fixed axis. Motion of a figure in a plane. Unilateral constraints ... 422
§ 3. Principle of virtual work. Virtual work. Unconstrained body. Body having a fixed point. Plane motion of a body. Body having a fixed axis. Body having a fixed axis of twist. Screw. Determination of stresses in the bars of a frame ... 434
§ 4. Determination of the position of equilibrium in a force field. Lagrange's multipliers ... 446
§ 5. Lagrange's generalized coordinates. Parameters of a system. Virtual displacements. Virtual work. Generalized forces. Conditions of equilibrium. Equilibrium in a potential field ... 451
CHAPTER X. DYNAMICS OF HOLONOMIC SYSTEMS [PDF] 
§ I. Holonomic systems. ... 466
§ 2. Non-holonomic systems. ... 467
§ 3. Virtual displacements. Point on a surface. Point on a curve. Examples. Systems of points. Generalized coordinates ... 468
§ 4. D'Alembert's principle. Equilibrium of forces. 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. Lagrange's equations in a potential field. Cyclic coordinates. Motion of a point on a surface of revolution. Spherical coordinates... 483
§ 8. Hamilton's canonical equations. Scleronomic systems ... 498
CHAPTER XI. VARIATIONAL PRINCIPLES OP MECHANICS [PDF] 
§ I. Variation without the variation of time. Variation of a function. Variation of an integral. Variation of a derivative. Variation of a compound function. Systems of points ... 504
§ 2. Hamilton's principle. Actual motion. Comparative motion. Hamilton's principle for natural coordinates. Hamilton's principle for generalized coordinates. Hamilton's principle in a potential field. Holonomo-scleronomic systems in a potential field... 512
§ 3. Variation with the variation of time. Variation of a function. Systems of points. Variation together with the variation of time of an integral ... 522
§ 4. Maupertuis' principle (of least action). Holder's transformation. More general form of Hamilton's principle. Maupertuis' principle ... 527
Appendix [PDF] 
Ordinary differential equations of the second order with constant coefficients ... 534
Index [PDF] 
Acknowledgements
We deeply thank Prof. dr. hab. med. Alina Filipowicz-Banach and the whole of family of Stefan Banach for their permission to post of all the works of Stefan Banach on this website. We also thank Prof. dr. hab. Stefan Jackowski, the Head of the Polish Mathematical Society, and Prof. dr. hab. Stanisław Janeczko, the Head of the Mathematical Institute of Polish Academy of Sciences, for their permission to post copies of the books published by the Polish Mathematical Society in the "Monografie matematyczne" serious on this website.

Questions or comments about this page can be sent to Emilia Jakimowicz or Adam Miranowicz. We would also appreciate every link from your pages to our Home Page of Stefan Banach.


File translated from TEX by TTHgold, version 4.00.
On 04 Jan 2012, 18:51.