Home Page of Stefan Banach
Home Page of Stefan Banach
by Stefan Banach
"Monografie Matematyczne" series, vol. 25, Polish
Mathematical Society, Warszawa-Wrocław, 1951, p. IV+546.
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.
Lwów, January, 1938.
CHAPTER I. THEORY OF VECTORS [PDF] |
§ 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
§ 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
§ 17. Central axis. Wrench ... 26
§ 18. Centre of parallel vectors ... 27
§ 19. Elementary transformations of a system ... 28
I. Operations on vectors.|
CHAPTER II. KINEMATICS OF A POINT
§ 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 ...
§ 8. Angular velocity and acceleration. Angular velocity vector
§ 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
I. Motion relative to a frame of reference.|
§ 12. Relation among coordinates.Motion along a holix. ... 52
§ 13. Relation among velocities ... 56
§ 14. Relations among accelerations. Acceleration of Coriolis ...
§ 15. Determination of relative motion. Motion relative to a
point. ... 65
II. Change of frame of reference.|
CHAPTER III. DYNAMICS OF A MATERIAL 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 ...
§ 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
§ 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
§ 16. Conditions for equilibrium in a force field. Stable
equilibrium. Dirichlet's theorem ... 118
I. Dynamics of an unconstrained 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
II. Dynamics of a constrained point.|
§ 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
III. Dynamics of relative motion.|
§ 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
CHAPTER IV. GEOMETRY OF MASSES [PDF] |
§ 6. Density. Calculation of mass. Material surface, material line
§ 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
II. Solids, surfaces and material lines.|
§ 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 ...
§ 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 V. SYSTEMS OF MATERIAL POINTS
§ 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
§ 8. Conditions of equilibrium. ... 207
§ 9. Reactions of bodies in contact. Normal and tangential
reactions. Supports. Centre of pressure. Reactions of a string...
§ 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
§ 12. Equilibrium of heavy supported bodies. Body supported at two
points. Body supported at n > 2 points ... 278
§ 13. Internal forces. ... 284
§ 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. ...
CHAPTER VI. STATICS OF A RIGID BODY
§ 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
§ 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
CHAPTER VII. KINEMATICS OF A RIGID BODY
§ 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. ...
§ 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 VIII. DYNAMICS OF A RIGID BODY
§ 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 ...
CHAPTER IX. PRINCIPLE OF VIRTUAL WORK
§ 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 X. DYNAMICS OF HOLONOMIC SYSTEMS
§ 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
Ordinary differential equations of the second order with constant
coefficients ... 534
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.
CHAPTER XI. VARIATIONAL PRINCIPLES OP MECHANICS
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