Mechanics

MECHANICS. This branch of applied mathematics deals with the motions of bodies; with the forces by which those motions are conditioned, and with the balance of forces on a body at rest. The word implies a connection with machinery (Gr. Anxavn) ; but this and other practical applications are to-day more commonly included under the heading of "applied mechanics," which covers such subjects as elasticity and the strength of materials, hydro mechanics and aerodynamics, mechanism, ballistics, etc. (qq.v.).

Theoretical mechanics, the foundation of all these subjects, may be divided into two closely related parts : dynamics, which is concerned with moving bodies, and statics, which treats of equilibrium, or rest. Dynamics—so-called because one aspect of the interaction between bodies, by which their motions are con ditioned, is the occurrence of what we recognize as force (Gr. bi)vailts)—may again be subdivided into kinematics, which deals with motion from the standpoint of measurement and precise de scription, and dynamics proper, which is concerned with causes, or "laws" of motion. Statics, the theory of balanced forces, can be established on foundations of its own, as an independent science ; but it is now customary to base it on the laws of dynamics, of which science it thus becomes a special branch. This procedure will be followed in this article, which attempts to present the essential features of the Newtonian scheme.

Speaking broadly, two standpoints are possible. The first pre sents dynamics as a science which has been constructed, by induc tion, on a basis of experiment. Corresponding with the axioms of Euclidean geometry (which "neither require, nor are capable of proof") we have, in dynamics, "laws of motion," e.g., a body which is not disturbed by force continues to move with uniform speed in a straight line. These "laws," it is claimed, can be put to the test of experiment ; and on them, step by step (as in the successive propositions of Euclid), is developed a system not only embracing in its scope all motions which occur in the material universe, but which has been proved by actual trial to maintain contact with that universe at every stage.

Difficulties in Verification of Laws.

It is recognized, of course, that practical limitations prevent an exact verification of any theoretical "law"; but a far more serious objection can he advanced against this presentation of dynamics, in that it involves an "argument in a circle": with whatever refinement the experiment is made, it cannot be interpreted without recourse to ideas which are themselves an essential part of the theory under examination. For example, if we seek to verify the "law" which

has been stated above, our first requirement is a body free from the action of force ; but no such body is available for test, because any body to which we can have access is subject to the earth's attraction. Therefore we must arrange that the attraction, since it cannot be eliminated, shall be neutralized, and this is attempted in what is known as "Attwood's machine": the body under test is connected, by a light string passing over a freely-running pulley, with a second body of equal weight; and under these conditions it is found that, started with any initial velocity, it retains that velocity almost unchanged. However quite apart from the fact that a small reduction in velocity (which may reasonably be attributed to friction) is always observed in any actual experi ment, there are difficulties in accepting this result as a proof of the "law" in question. We have found that a body moves with (substantially) uniform speed and direction; but it is not a body on which no force is acting, and without recourse to the prin ciples of dynamics (themselves dependent on the law) we have no grounds for asserting that the forces which act upon it do in fact neutralize one another.

Dynamics as an Abstract Science.

Difficulties of this kind are avoided if we adopt the alternative standpoint, according to which dynamics, as a purely abstract science, may legitimately be founded upon any set of initial assumptions (or "laws of motion") which is convenient and not self-contradictory. In con structing this theoretical system, we are under no obligation to verify that contact with the actual universe is maintained at every stage ; for the system is concerned with the motions of purely ideal bodies whose properties are postulated in its funda mental assumptions, and provided that it is developed by logical processes, according with the accepted principles of mathematics, its conclusions will be valid consequences of those assumptions. Whether it will lead to results having any correspondence with the observed motions of actual bodies is an entirely separate ques tion, which must be decided a posteriori, by comparing these results with experiment. But the available tests apply only to the system as a whole : we cannot devise an experiment which will verify any one of its assumptions apart from the rest.