# A treatise on dynamics - Besant

A TREATISE ON DYNAMICS

BY W. H. BESANT

CAMBRIDGE; DEIGHTON, BELL AND CO.; 1893

A treatise on dynamics

PREFACE

The object of the present treatise is to introduce the mathematical student to some of the earlier and easier branches of Phoronomy, the purely geometrical science of motion, and of Kinetics, the science which deals with the action of forces in producing motion, or changes of motion, in a body or system of bodies.

In the chapter allotted to Phoronomy, I have deduced the expressions for velocities and accelerations, as far as possible, from the definitions and axioms of the subject.

In the applications to Kinetics, or, in other words, in the combination of these expressions with the Laws of Motion, for the determination of the motion of a particle or of a system, I have adopted the same plan of operations.

I have assumed, and made free use of, the methods of Analysis, for the performance and simplification of the requisite calculations.

The methods employed, and the order of thought which is followed, are those which during my experience as a teacher I have found to be most effective in the elucidation and development of the ideas of Phoronomy and Kinetics.

The majority of students do not easily or rapidly absorb general ideas, and they are most effectively assisted by the gradual development of a subject through simple cases, and illustrative examples.

With this view I have endeavoured to explain the application of the Laws of Motion to the determination of the motion of a particle and of systems of particles, commencing with easy cases, and leading up to a few of the interesting and important cases of the motions of bodies and of systems of bodies.

My especial object has been to illustrate the direct application of the Laws of Motion, and thereby to produce a treatise of an elementary character, but of Educational utility to the student who is commencing the study of theoretical Kinetics.

In particular the chapter on the motion of a particle in three dimensions has been considerably expanded, and two new chapters have been added, one on disturbed elliptic motion and the other on the Lagrange equations.

Chapters XV and XVI are an expansion and rearrangement of chapter XIV of the first edition.

In a letter which was published in Nature on March 17, 1892, I gave my reasons, philological and historical, for employing the word Phoronomy instead of the word Kinematics.

As a matter of philology the word Phoronomy represents the ideas of pure motion, without regard to causation, more correctly than the word Kinematics.

As a matter of history the word was first used by Hermann, whose treatise, Phoronomia, was published in 1716 at Amsterdam. Hermann however employed the word to represent the general science of motion, including the action of forces.

I venture to hope that the explanations and illustrations of the text, and the numerous examples appended to the several chapters, will be of assistance to the student in mastering the elementary ideas of the subject, and pave the way for the consideration of the higher branches and the more difficult problems of the great science of Dynamics.

W. H. BESANT.

CONTENTS.

CHAPTER I
Introductory

CHAPTER II
Differential Equations

CHAPTER III
Phoronomy
Examples

CHAPTER IV
Laws of Motion

CHAPTER V
Rectilinear Motion
Examples

CHAPTER VI
Accelerations Parallel to Co-ordinate Axes
Examples

CHAPTER VII
Examples

CHAPTER VIII
Tangential and Normal Accelerations
Examples

CHAPTER IX
Disturbed Elliptic Motion

CHAPTER X
Motion of a Particle in Three Dimensions
Examples

CHAPTER XI
The Hodograph and the Brachistochrone
Examples

CHAPTER XII
Motion of Two Particles acting on each other
Examples

CHAPTER XIII
Energy and Momentum
Examples

CHAPTER XIV
Equations of Motion
Examples

CHAPTER XV
Motion of a System in Three Dimensions
Examples

CHAPTER XVI
Motion of a Top, Motion under no forces, Motion of rolling Disc, Euler's Equations
Examples

CHAPTER XVII
The Lagrange Equations
Examples

DYNAMICS

CHAPTER I

1. The problems usually discussed under this head are those which relate to the geometrical connections between given motions, or given kinds of motion, and those which relate to the action of forces, and the motions and changes of motion produced by forces.

The former belong to pure science, and deal with the geometry of motion, a branch of mathematics to which the name Kinematics was applied by Ampere.

"We shall however employ the word Phoronomy to represent the purely geometrical science of motion in the abstract.

Strictly speaking the word Dynamics includes Statics, the discussion of the equilibrium or balancing of forces, and Kinetics, the discussion of the effects of forces on the motion of bodies.

Mechanism, including such problems as result from considering trains of wheel- work or any connected machinery, is really a branch of Phoronomy.

Some writers employ the word Kinematics to represent what is commonly called Mechanism.

To Kinetics belong the consideration of the forces setting such machinery in motion, or keeping it in motion, the problems of Physical Astronomy, and others of important practical application.

We shall commence by a development of the formulae of Phoronomy, and afterwards proceed to consider the application of the formulae, and of the Laws of Motion, or Laws of Force, to the determination of the motion of a particle, and of a system of particles, produced by the action of given forces, or, conversely, of the forces required to produce given motions. The idea of a particle, or of a material point, capable of being set in motion, or of having its motion affected, by the action of force, is a mathematical abstraction leading to the simplest forms of Kinetics. The determination of the motions of the bodies constituting the Solar System belongs to this class in virtue of the facts that the Planetary Bodies are nearly spherical in form, and that their dimensions are very small in comparison with their distances from each other and from the Sun.

Moreover the mathematical idea of a solid body is that of a system of particles, and the discussion of the motion of a single particle therefore naturally precedes the discussion of the motion of a body or system of particles.

It will be seen that Newton's Laws of Motion connect the action of a force on a particle with the accelerations produced, and lead to the formation of differential equations, the integration of which gives the solution of the problem of determining the motion.

It will appear further that Newton's Laws are sufficient for the determination of the motion of a system of particles or bodies, whether rigidly connected or not, and lead, in a similar manner, to systems of differential equations containing in their solution the motions of the body, or of the various bodies of the system.