Newtonian mechanics

“ Hence, since the laws of Newtonian mechanics are based on inertial systems, it follows that all such systems are equivalent for the description of the laws of mechanics: no one system amongst them is unique, and we cannot define absolute motion or rest; any systems moving with mutual rectilinear uniform motion may be regarded as being at rest. Mathematically, this means that the laws of mechanics remain unchanged in form for any transformation from one set of inertial axes to another. ”

“ In short, the principles of conservation endured only when we enlarged our understanding of mass, momentum and energy; and Newtonian mechanics had to be amended as a result of Einstein’s discoveries in the special theory of relativity.
Let us now study the problems of conservation in the general theory when we make use not merely of a Galilean frame of reference, but of any frame whatever, hence when our space is permeated with a field of inertial forces or, again, with a field of gravitation. We shall discover once more that conservation is required by the general theory ”

“ We have shown how observers stationed on systems in motion relative to one another have been able to preserve their fundamental principles of mechanics only by adopting certain novel conclusions. These conclusions are self-consistent; in the one case where they have been tested they are in accord with experiment; and they enable us to save all the fundamental physical concepts which have been found useful in the past. We have, however, considered primarily only systems which are initially in uniform relative motion. ”

“ Let us now return to Newton’s solution of the problem, since it constitutes the embodiment of classical mechanics. As we have seen, Newton, by positing his postulate of isolation, rendered inevitable a belief in some real absolute medium, space or the ether, from which rotation and acceleration would derive a real meaning. But here we must recall that mechanical experiments, though clearly differentiating a Galilean frame from a non-Galilean frame, were totally incapable of differentiating one Galilean frame from another. ”

“ But although physical measurements become inevitable as soon as we wish to obtain a concrete definition, an objective criterion of equal durations, we are able to proceed in our mechanical deductions in a purely mathematical way without further appeal to experiment. The principle of inertia, together with the other fundamental principles of mechanics, enables us, therefore, to place mechanics on a rigorous mathematical basis, and rational mechanics is the result. ”

“ Commencing with the generalization that " a mechanical force, or the cause of a mechanical motion, we know to be in general the condition of a difference of pressure," he infers, "hence it appears that if a general mechanical theory is possible, the ultimate property of matter must be conceived to be a mutual repulsion of its parts, and the indubitable Newtonian law of universal attraction be deduced herefrom, under the actual conditions of the world. ”

“ But it is obvious that we have no right to assume that Newton’s law or those of mechanics are above all suspicion. The laws of physical science present no a priori inevitableness; and our sole reason for accepting them is because they appear to be borne out to a high degree of approximation when we compare their rational consequences with the results of observation. Approximation is thus all we can aspire to in physical science; and even classical scientists recognised the possibility that more precise observation might prove Newton’s law and those of mechanics to stand in need of correction. ”

“ Chemistry here comes to the aid of mechanics, and we may say boldly that that matter, which is identical with ours, is subject, like it, to the laws of gravitation. Certainly, when Newton decomposed a beam of white light, and laid the first basis of the theory of the spectrum, he had not the slightest suspicion that his law of gravitation would, at a later period, find in it wings to carry it into regions where all measurement ceases and all calculation is powerless. ”

“ This part of mechanics was cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power) , no otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as the mathematical principles if philosophy ”

“ The greatest advances of science have always consisted in some successful formulation, in clear, abstract, and communicable terms, of what was instinctively known long before, and of thus making it the permanent property of humanity. By Newton's principle of the equality of pressure and counterpressure, whose truth all before him had felt, but which no predecessor had abstractly formulated, mechanics was placed by a single stroke on a higher level. Our statement might also be historically justified by examples from the scientific labors of Stevinus, S. Carnot, Faraday, J. R. Mayer, and others. ”

“ Now, a general theorem of mechanics proves that whenever a body is moving under the attraction of a central body, the law of areas is always verified; whence we must conclude that the sun could never be considered as rotating round the earth under the sole action of the earth’s attraction. Supplementary forces would have had to be brought into play, but their introduction would have been so arbitrary, and the mathematical difficulties so great that even Newton might have been nonplussed ”

“ This accordingly was done, by the founders of mechanics, both for the cases of the equilibrium and of motion. Newton’s attempt to solve the problem of the figure of the earth, supposing it fluid, is the first example of such an investigation: and this solution rested upon principles which we have already explained, applied with the skill and sagacity which distinguished all that Newton did. ”

“ If we confine the application of the theory to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the co-ordinate system with velocities which are small compared with the velocity of light, we then obtain as a first approximation the Newtonian theory. Thus the latter theory is obtained here without any particular assumption, whereas Newton had to introduce the hypothesis that the force of attraction between mutually attracting material points is inversely proportional to the square of the distance between them. ”

“ Newton’s law of gravitation stated that the attractions exerted between bodies depended solely on their masses and mutual distances; the conditions of relative motion or rest of the bodies being considered irrelevant. But if Mach’s ideas are accepted, we must assume that in addition to the ordinary Newtonian attraction, a supplementary field of attraction, hitherto unsuspected, should arise if the bodies were in a state of relative acceleration, as in the case of the stars rotating round the earth, or the planets moving round the sun. ”

“ But to treat of Mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law of motion. For if we estimate the action of the agent from its force and velocity conjunctly; and likewise the reaction of the impediment conjunctly from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts; the action and re-action in the use of all sorts of Machines will be found always equal to one another. ”

“ If a body were moving with respect to our frame of reference with a relative velocity, but with no relative acceleration, we should be justified according to the Newtonian principle in maintaining that it would be impossible to decide by mechanical experiments whether it was we who were moving towards the body or the body that was moving towards us. ”

“ In his Preface, Newton says, “Rational Mechanics must be the science of the Motions which result from any Forces, and of the Forces which are required for any Motions, accurately propounded and demonstrated. For many things induce me to suspect, that all natural phenomena may depend upon some Forces by which the particles of bodies are either drawn towards each other, and cohere, or repel and recede from each other: and these Forces being hitherto unknown, philosophers have pursued their researches in vain. ”

“ All doubts about classical mechanics set in at the very commencement with the formulation of the law which Newton places at its head, the formulation of the law of inertia.
As has already been emphasized on page 31, the assertion that a point-mass which is left to itself moves with uniform velocity in a straight line, omits all reference to a definite co-ordinate system. An insurmountable difficulty here arises: Nature gives us actually no co-ordinate system, with reference to which a uniform rectilinear motion would be possible. ”

“ We could reconstruct all mechanics by attributing different values to all the masses. This new mechanics would not be in contradiction either with experience or with the general principles of dynamics (principle of inertia, proportionality of forces to masses and to accelerations, equality of action and reaction, rectilinear and uniform motion of the center of gravity, principle of areas) . ”

“ The Principia can hardly be said to contain any new inductive discovery respecting the principles of mechanics; for though Newton’s Axioms or Laws of Motion which stand at the beginning of the book, are a much clearer and more general statement of the grounds of Mechanics than had yet appeared, they do not involve any doctrines which had not been previously stated or taken for granted by other mathematicians. ”

“ One plan is to assume that any body whatever may be treated as if it were composed of material particles, i.e. mathematical points endowed with inertia coefficients, separated by finite intervals, and acting on one another with forces in the lines joining them subject to the law of equality of action and reaction. In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered. On this basis we can predicate the principles of linear and angular momentum, as in § 15. ”

“ If we are concerned solely with the facts of planetary motion and mechanics known to Newton, then Newton’s laws of mechanics and his law of gravitation are by far the simpler. But if we supplement the facts known to him with those more recently discovered, Einstein’s synthesis has the advantage, for Newton’s co-ordination would necessitate a number of additional hypotheses ad hoc. In the same way, so far as our everyday experience is concerned, a non-rotating earth is the simpler solution, whereas, when the planetary motions are taken into consideration, the reverse holds true. ”

“ To practical Mechanics all the manual arts belong, from which Mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that Mechanics is so distinguished from Geometry, that what is perfectly accurate is called Geometrical, what is less so is called Mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy, is an imperfect Mechanic, and if any could work with perfect accuracy, he would be the most perfect Mechanic of all. ”

“ And since similar, equal, and equally swift bodies, moving through mediums of the same density, whose particles do not fly from each other mutually, will strike against an equal quantity of matter in equal times, whether the particles of which the medium consists be more and smaller, or fewer and greater, and therefore impress on that matter an equal quantity of motion, and in return (by the 3d Law of Motion) suffer an equal re-action from the same, that is, are equally resisted ”

“ Thus Newton's first law, for instance, would have to be restated somewhat as follows: No physical body is ever in a state of rest or uniform motion in a straight line, unless its natural condition is interfered with by the particular action of some force. ”

“ That every particle of matter in the universe attracts every other particle with a force that varies directly with the mass of the particles and inversely as the squares of their mutual distance. Newton did not vault at once to the full expression of this law, though he had formulated it fully before he gave the results of his investigations to the world. ”

“ Newton opined that an object moves because of external forces on it rather than by forces internal to the object. He connected the concepts of force and acceleration with a new concept: mass. He found that the acceleration of a body by a force is inversely proportional to its mass, and formulated the equation that force equals mass time acceleration. Another law was his principle of inertia that any body, in so far as it is able, continues its state either of rest or in uniform, rectilinear motion. ”

“ For in those Propositions, instead of the uniform resistance made to an ascending body arising from its gravity, one may substitute the uniform resistance which arises from the tenacity of the medium, when the body moves by its vis insita alone; and when the body ascends in a right line, add this uniform resistance to the force of gravity, and subduct it when the body descends in a right line. One might also go on to the motion of bodies which are resisted in part uniformly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity. ”