Vectorial Mechanics
Encyclopedia
Vectorial Mechanics is a book on vector manipulation (i.e., vector methods) by Edward Arthur Milne
, a highly decorated (e.g., James Scott Prize Lectureship
) British astrophysicist and mathematician. Milne states that the text was due to conversations (circa 1924) with his then-colleague and erstwhile teacher Sydney Chapman
who viewed vectors not merely as a pretty toy
but as a powerful weapon
of applied mathematics
. Milne states that he did not at first believe Chapman, holding on to the idea that "vectors were like a pocket-rule, which needs to be unfolded before it can be applied and used." In time, however, Milne convinces himself that Chapman was right.
, particle dynamics, types of particle motion, dynamics of systems of particles, rigid bodies in motion, dynamics of rigid bodies, motion of a rigid body about its center of mass
, gyrostatic problems, and impulsive motion.
G.J.Whitrow
:
Daniel C. Lewis:
Arthur Milne
Edward Arthur Milne FRS was a British astrophysicist and mathematician.- Biography :Milne was born in Hull, Yorkshire, England...
, a highly decorated (e.g., James Scott Prize Lectureship
James Scott Prize Lectureship
The James Scott Prize Lectureship is given every four years by the Royal Society of Edinburgh for a lecture on the fundamental concepts of Natural Philosophy. The prize was established in 1918 as a memorial to James Scott by trustees of his estate....
) British astrophysicist and mathematician. Milne states that the text was due to conversations (circa 1924) with his then-colleague and erstwhile teacher Sydney Chapman
Sydney Chapman (astronomer)
Sydney Chapman FRS was a British mathematician and geophysicist. His work on the kinetic theory of gases, solar-terrestrial physics, and the Earth's ozone layer has inspired a broad range of research over many decades....
who viewed vectors not merely as a pretty toy
Toy weapon
Toy weapons are toys that mimic real weapons, but are designed to be fun for children to play with and not dangerous.-Types of toy weapons:Some are essentially similar to the real thing, but less powerful. Weapons for cutting and stabbing have dull blades usually in plastic. Weapons formerly made...
but as a powerful weapon
Weapon
A weapon, arm, or armament is a tool or instrument used with the aim of causing damage or harm to living beings or artificial structures or systems...
of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
. Milne states that he did not at first believe Chapman, holding on to the idea that "vectors were like a pocket-rule, which needs to be unfolded before it can be applied and used." In time, however, Milne convinces himself that Chapman was right.
Summary
Vectorial Mechanics has 18 chapters grouped into 3 parts. Part I is on vector algebra including chapters on a definition of a vector, products of vectors, elementary tensor analysis, and integral theorems. Part II is on systems of line vectors including chapters on line co-ordinates, systems of line vectors, statics of rigid bodies, the displacement of a rigid body, and the work of a system of line vectors. Part III is on dynamics including kinematicsKinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
, particle dynamics, types of particle motion, dynamics of systems of particles, rigid bodies in motion, dynamics of rigid bodies, motion of a rigid body about its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, gyrostatic problems, and impulsive motion.
Summary of reviews
There were significant reviews given near the time of original publication.G.J.Whitrow
Gerald James Whitrow
Gerald James Whitrow was a British mathematician, cosmologist and science historian.After completing school at Christ's Hospital, he obtained a scholarship at Christ Church, Oxford in 1930, earning his first degree in 1933, the MA in 1937, and the PhD in 1939...
:
Although many books have been published in recent years in which vector and tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
methods are used for solving problems in geometry and mathematical physicsMathematical physicsMathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, there has been a lack of first-class treatises which explain the methods in full detail and are nevertheless suitable for the undergraduate student. In applied mathematics no book has appeared till now which is comparable with Hardy's Pure MathematicsA Course of Pure MathematicsA Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions and several reprints. It is now out of copyright in UK and is downloadable from...
. ... Just as in Hardy's classic, a new note is struck at the very start: a precise definition is given of the concept "free vector", analogous to the Frege-Russell definition of "cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
." According to Milne, a free vector is the class of all its representations, a typical representation being defined in the customary manner. From a pedagogic point of view, however, the reviewer wonders whether it might have been better to draw attention at this early stage to a concrete instance of a free vector. The student familiar with physical concepts which have magnitude and position, but not direction, should be made to realise from the very beginning that the free vector is not merely "fundamental in discussing systems of position vectors and systems of line-vectors", but occurs naturally in its own right, as there are physical concepts which have magnitude and direction but not position, e.g. the coupleCouple (mechanics)In mechanics, a couple is a system of forces with a resultant moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass.The resultant moment of a couple is called...
in statics, and the angular velocityAngular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
of a rigid bodyRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
. Although the necessary existence theorems must be established at a later stage, and Milne's rigorous proofs are particularly welcome, there is no reason why some instances of free vectors should not be mentioned at this point."
Daniel C. Lewis:
The reviewer has long felt that the role of vector analysis in mechanicsMechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
has been much overemphasized. It is true that the fundamental equations of motion in their various forms, especially in the case of rigid bodies, can be derived with greatest economy of thought by use of vectors (assuming that the requisite technique has already been developed); but once the equations have been set up, the usual procedure is to drop vector methods in their solution. If this position can be successfully refuted, this has been done in the present work, the most novel feature of which is to solve the vector differential equations by vector methods without ever writing down the corresponding scalar differential equations obtained by taking components. The author has certainly been successful in showing that this can be done in fairly simple, though nontrivial, cases. To give an example of a definitely nontrivial problem solved in this way, one might mention the nonholonomic problem afforded by the motion of a sphere rolling on a rough inclined planeInclined planeThe inclined plane is one of the original six simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. By moving an object up an inclined plane rather than completely vertical, the amount of force required is reduced, at the expense of increasing the...
or on a rough spherical surface. The author's methods are interesting and aesthetically satisfying and therefore deserve the widest publication even if they partake of the nature of a tour de force.