The fundamental equations of dynamics and its main coördinate systems vectorially treated and illustrated from rigid dynamics

Preface

The day has clearly passed when any comprehensive presentation of all dynamics could be compressed and unified within the compass of one moderate volume of homogeneous plan. The established connections of dynamical reasoning with other fields in physics are of increasing number and closeness, as furnishing for them strongly rooted sequences in their interpretative trains of thought and linking together what would else have continued to stand separate. And that relation has reacted powerfully in modern times upon dynamics itself, perpetually enriching its substance, yet at the same time introducing within it certain sharpening differences that are stamped upon it by the type of use for which preparation is being made. These in fact modify superficially the modes of expression and their tone, and shift their own emphasis through a range that brings about what is in effect a subdivision of territory and an acknowledgment of practically diverse interests. It is in response to the situation which has been thus unfolding, and in conformity with its drift toward manifold adaptations, that special treatises have been rendered available whose measure of unquestioned excellence and authority would make superfluous an attempt to replace any such unit with a marked improvement upon it.

But undoubtedly these differentiations founded in divergencies and inevitably expressing them in some degree, are entailing a corresponding need and demand to offset them with a broadening survey of the common foundation and of the common stock of resources. And with that end in view another treatment of dynamics finds a place for itself and holds it for special service. This will propose to state with catholic inclusiveness the principles iv Fundamental Equations of Dynamics

that lay out and direct all the main lines of use, and to anticipate at their common source, as it were, the preferred methods and forms that are characteristic of various provinces.

On this side also reasonable requirements for the immediate future have been satisfied up to a definitely recognizable point. For works on abstract dynamics are at hand to help, whose number and quality have left no fair opening for renewed exposition, that could indeed scarcely attain excellence without duplicating them. In the same proportion, however, that their requisite perspective has grown, until it involves truly panoramic sweep, its due scope must cease to be secured except from a distance that expunges most details and spares only landmarks of the bolder outlines. And under the urgent pressure to condense in order to avoid neglecting and yet not become too voluminous in summarizing completely, to keep even pace with widening outlook, this view of dynamics cannot but endure the attendant risks of abstractness. Because it must lean in building toward great reliance upon the formal aid of mathematics, perforce the physical coloring will fade and the bonds with experimental reasoning be loosened. The stated results are progressively less likely to comprise what is charged with tentative quality and is held with the candidly provisional acceptance proper to inductive method.

For a student devoted to physical science though, as the gifted mathematicians Poincare and Maxwell have been anxiously insistent that he should be aware, there are lurking elements of danger in magnifying a bare logical skeleton as a goal, and in spending best effort upon articulating it. It is a misguidance apt to control into rigidity thought which can scarcely prove worthily fruitful unless it is maintained plastic. There is a plain sense in which dependence upon clarity of demonstration in terms of mathematical brevity and rigor may operate as a defect; and that severe pruning which suppresses all but defini-

Introduction v

live advance may mislead. There is a season for mitigating the austerity of algebra and daring to become discursive, for relaxing the ambition that is steadied to attain command of abstract principles on their highest level and for pausing in reflective examination of their genesis and their setting. Truly it would sterilize action to incline thus always; but never to turn aside from the more arduous pursuit tends to dissipate that atmosphere for dynamics which has given it life.

At the other extreme are found the practical temperaments, looking for tools with which to undertake their special tasks, and largely unmindful of the processes by which those have been shaped and of the far-reaching equipment in which their function is but one part, if only a particular routine can be adequately served or intelligently mastered. And this more empirical frame of mind that springs from absorption in monopolizing pursuits can be fostered and strengthened by the sheer difficulties in external form that are impressed upon abstract dynamics by the tendencies that have just been referred to, and by the air of remoteness from things material and mundane which that treatment, if uncorrected, confers. Unless it can be halted, therefore, a movement toward disintegration which must be coped with will confront the cultivators of dynamics that derives a backing also from other circumstances of the present situation.

The lifting of technical science to a better plane, where the habitual facing of new problems under the illumination of theoretical insight is coming to prevail, creates a demand in all the fundamental sciences that is a modern appeal. It has been incorporated into fixed plans of preparation for normal careers in active life to accomplish those things which were formerly undertaken with dominating inclination by minds self-selected through their special gifts. There must be, then, in the methods of presentation and in the execution of them, some recognition vi Fundamental Equations of Dynamics

of a constituency that is at once larger, less homogeneous, and more in need of aid. In a restricted sense of the word, there is a summons to popularize the abstruser sciences, and among them dynamics, with a design to favor their assimilation by students at an earlier stage. This will make concessions in view of hindrances inherent in the subject-matter, and allowance for faculties of comparison and of analytic judgment not yet ripened into full command of all resources.

There is some element in the immediate need that is due to passing a transition and that will be lost in a newly adjusted order; for it has appeared from manifold experience what marvels can be wrought by tradition in giving easy currency to scientific doctrine. Moreover, the obstacles that loomed larger by mere novelty suffer genuine reduction by more lucid statement. An older generation arrived but gradually at an understanding of the principle in conservation of energy, and caught the advantage and power of absolute measurements first in glimpses. Yet they have lived to find those unfamiliar ideas adopted among the smoothly working formulas of unquestioned truth. So it will not pass the limits of a reasonable anticipation to forecast how the younger generation of todaj^ can move at ease in their maturity among bold concepts that were obscure when imperfectly grasped. Nevertheless, as the call now is, so must the answer be given.

Every aspect of the thoughts here put down is framed in a personal experience: the profit from quickening perception and appreciation for the nexus between sharply generalized ideas and their narrower origins; the benefit of laying steppingstones gauged to a student's stride; the reward of implanting human interest within the routine of an industrial calling; also the moral gain through confirming intellectual honesty under a sustained demand for actual comprehension of what one is challenged to attack among the papers rated as classics, or in

Introduction vii

judging and sifting recent work. Aiding to scent difficulties first and then to overcome them fits the processes of the average mind, where the stronger talent can walk self-guided.

The present enterprise was born of the foregoing considerations in so far as they dictated its material and the ends for which that was offered in gradual accumulation during many years and under the influence of contact with students of varied purpose. It renounces from the outset all claim to be systematically conceived; it is content with a circling return from one point and another to a core of ideas that are worth reviewing in their various aspects because they are central. In their nature being a supplement to standard books that differ in type from each other, and offering themselves in flexible continuation of an elementary stage with unsettled achievement, these selected discussions cannot escape being judged fragmentary by some, redundant by others. But their spirit and their general aim are built upon ascertained failure to acquire elsewhere a just comprehension of several matters here made prominent and perhaps in some degree originally presented.

This kernel of intention in the subject-matter gathered for these chapters lends to them, it may be claimed legitimately, something of peculiar appropriateness for the circumstances of their publication. On the occasion to be celebrated it seems particularly pertinent that there should be recorded in some permanent form the working of those influences which our University has not withheld from her graduates, to nourish in them a living root of independent thinking and of unflinching thoroughness without which constructive scholarship cannot exist.

CONTENTS CHAPTER I

Page

Chapter I

1. Only sciences that have attained a certain ripeness of strongly rooted development have been found capable of combining a vigorous and progressive activity at their working frontier for advance with reflective examination of their deeper foundations and their general method. The activity is aggressive in devising novel attack upon enlarging material, while reflection upon what has already become standard must go with recasting it to meet modified demands. This situation has been prominently realized in the case of dynamics, whose stirrings to selfcriticism have been evermore spurred by the interactions with mathematics and astronomy, its closer neighbors, at the same time that its field was broadening to permeate and harmonize the greater part of physics. A large net gain of helpful stimulus from common aim must be allowed here, reenforcing the vigor from rapid growth, though there have been some dangers for dynamics to avoid, such as becoming infected with the more formal and abstract spirit of mathematics, or underrating its own basis in phenomena by acquiescing too generously in philosophy's rating for empirical science. It is a fitting preliminary to our immediate purpose to touch upon one or two such reactions between influences from without and from within; in part because the inquiries that were provoked, though prolonged through fifty years or more with acuteness and tenacity, have left practically unshaken the external forms of quantitative expression, at least. This is no sign, however, that dynamics is stationary and stereotyped; but only a reassuring fact to beget confidence in the fabric of the science. The subtle and less obtrusive changes must not be forgotten, that have clarified the concepts and infused into them added significance by revised interpretation. Reading the prospects of the imminent future, too, rouses the expectation that what has been will continue to be, while dynamics is adapting itself to a wider scheme of connections and to a more accurate insight into its own doctrine or theory.

It is indeed an astonishing testimony to the happy strokes of genius in the founders of mechanics that force, impulse, work, momentum and kinetic energy still exhaust the primary needs, though the broader scope of dynamics now covers the chain of transformations in which mechanical energy is only one link. And it confirms our belief in the vital and definitive appropriateness of those quantities to find them retained essentially by those who are trying out another body of principles that might be substituted entirely or in part for the Newtonian mechanics. Meanwhile the equations of motion have not been superseded, yet they date from the seventeenth century; the notable advances due to d'Alembert, Euler and Lagrange in the eighteenth century, and to Hamilton in 1835, offer still the foundations upon which we build. But this introduction would outline a one-sided and misleading picture of mere static stability unless it used its warrant in bringing to supplementary notice three strands that have been woven into dynamics more recently, to alter in some degree its texture and to influence its emphasis. We shall next attempt to dispose of these in all proper brevity.

2. Under the first label energetics we are called upon to chronicle a strong movement that sought to enhance the prestige that energy in its various forms had already gained by the rapidly successful campaign about the middle of the nineteenth century.^ This tendency was an almost inevitable accompaniment of that dominating relation to physical processes which conservation of energy as a conceded central principle had justified beyond cavil.

^ See Note 1. Refer to collected notes following Chapter IV.

But the more pronounced utterances about energy overshot the mark in their zeal, and sought to exalt it in rank as the one dynamical quantity to which the rest should be held auxiliary, and upon which they should be based mathematically. Then the series — kinetic energy, momentum, force, mass — was to be unfolded out of its first term by divisions ; and violent extremists were heard, even condemning force as a superfluous concept, refusing to associate it directly with our muscular sense, or to recognize it as an alternative point of departure yielding momentum and other quantities by multiplications. Of course deliberate minds looked askance at a professedly universal point of view that would exclude, save at the cost of an artificial device, such important elements as constraints that do no work. Common sense declined to cripple our assault upon problems for doctrinaire reasons that would bar and mark for disuse certain highways of approach, but it seized the chance instead to enrich and strengthen dynamics by wisely adopting the suggestion to exploit more completely the relations that energy specially furnishes, and to incorporate them among its resources and methods. After abating its flare of exuberance, the saner forces behind the reconstruction that was advocated have been harnessed and made contributory to a real advance that grafts new upon old, and embraces whatever proved advantage attaches to all reasonable points of view, with the object of reducing finally their oppositions and fitting them in place within a more comprehensive survey.

What is patent to read in the example of energetics should in prudence be made further to bear fruit; since judging historically, any new burst of reform spirit will be likely to repeat the main features of its lesson. An old and thoroughly tested science especially will less easily break the continuity of its course, though it is always responsively ready to swerve under every fresh impulse to amendment by discovery. So the matters offered recently under the caption relativity are surely giving to dynamics a wider sweep of horizon; but there too, when the permanent benefit accruing has been sifted out, the residue will probably prove more moderate than the tone of radical spokesmen has been implying while the sensation of novelty was strongest.^

3. It has been remarked often that Newton's three laws of motion taken by themselves give a bias toward concentrating attention upon momentum, and upon force exclusively as its time-derivative, with a comparative neglect of the counterpart in work and its relation to force. The restoration of balance began at once however, and soon the principle of vis viva was added and recognized as complementary on a level footing to Newton's second law. The equivalents of what are now known as the impulse equation and the work equation were established firmly and put to use. The readjustment thus begun was continued by steps as their desirableness was felt until with the ripeness of time it culminated, we may say, in the proposals that form the nucleus of what we call energetics. It will be profitable to expand that thought and mention some chief sources of the need to follow that line, or what gain has been found in doing so.

In rudimentary shape the idea of conservation of energy had emerged early; the histories are apt to date it from the method invented by Huyghens for the treatment of the pendulum. And so soon as the formal step had been taken in addition, that set apart under the heading potential energy the work of weight and of gravitation, because it can be anticipated by advance calculation exactly and with full security, the invariance of mechanical energy under the play of these forces when thus expressed, or its conservation within these narrower limits, became a demonstrable corollary of fundamental definitions.

The discovered inclusion of electric and magnetic attractions or repulsions under the same differentially applied law of inverse square that is characteristic of gravitation made natural the extension of potential energy as a statement of securely anticipated work to the field of those actions as well. And a large group of valuable mathematical consequences was accumulated which remain classic and which accompany the law^ of inverse square wherever it may lead, retaining their validity with only slight changes of detail.

These developments are controlled to a great extent by the idea of energy, and they must have built up a general perception of its power. The invariance of energy was fitted more completely for use as a principle, wherever its mechanical forms alone enter which we distinguish as kinetic and potential, when Gauss had evolved that plan of so-called absolute measure which has furnished us with the centimeter-gram-second system. He certainly consolidated into unity all sources of ponderomotive force in the several fields where a potential had been recognized. Of course we discriminate between this stage and the conservation of energy under all its transformations to which the period of Mayer, Joule and their coworkers attained. The earlier halting-place behind distinct limitations of scope left matters besides with a formal content only, in the sense that no questions were raised and squarely faced that looked toward localizing the latent energy and investigating the possible mechanism by which a medium might hold it in storage. This formal mathematics centered on the fact that the work done within a conservative system and between the same terminal configurations does not depend upon the particular paths connecting them. It is a strikingly significant exhibition of that quasi-neutrality that is now one salient and accepted feature in the procedure of energetics that so much of solid and permanent accomplishment was possible while certain vital issues were evaded, and without being compelled to register even a tentative decision upon them. That non-committal attitude towards much else as subsidiary, provided always that the gains and losses of energy for the system under consideration can be made to balance, has often been employed to turn the flank of obstacles and has been in that respect an element of strength. Or it leaves us in the lurch weakly, we might say about other occasions where we have stood in need of some crucial test between alternatives, and have found but a dumb oracle.

4. The next important advance was then timely and specially fruitful in giving life and deeper meaning to what had been in these directions more a superficial form; and at the same time in moving forward beyond the previous stopping-place to expand the range of dynamical ideas. ^ It is Maxwell who is credited with initiating these contributions by treating dynamically new aspects of electromagnetic phenomena. He took bold and novel ground by outlining his provisional basis for an electromagnetic theory of light that converted a colorless temporary vanishing of energy into a definite and plausible plan for its storage in a medium. In achieving this change of front he brought three lines of thought to a convergence-point; for besides the researches of Faraday and those that identified quantitatively the many transformations of energy, he utilized more fully than his predecessors had dared the possibilities that the earlier dynamics had done much toward making ready to his hand. It is this third element perhaps that, marks most strongly for us the threshold of the new enterprise upon which dynamics will hereafter be engaged, in whose tasks we can find a union in just proportion of imaginative speculation with mastery of the mathematical instruments and with the candid policy of energetics to preserve an open mind and a suspended judgment in the face of undecided questions.

Maxwell was a pioneer in prolonging with new purpose the sequence upon which d'Alembert set out, and which Lagrange continued, beyond the point at which the latter paused after recording notable progress. What those earlier men had done with the discovery of virtual work as a basis for developing mechanics remained to be restated for dynamics, and adapted to a more inclusive command of energy transformations. Among other things this has given us an enlarged interpretation of older terms. We are ready to view a conservative system as one whose energy processes are reversible: that is, energy of any form being put in, it can be restored without loss, in the same form or in some other. We have learned to group fair analogues of kinetic and of potential energy for a system thus conservative according to one defensible test. Potential forms of energy will be found resilient as the original examples are; that is, they will exhaust themselves automatically, under the conditions of the particular combination, unless the corresponding transformation is prevented actively. But in order to be coordinated with kinetic energy on the other hand, the passive quality must be in evidence that requires some decisive intervention for the passage into other forms. This trend toward assigning wider meaning to dynamical concepts has given us further generalized force as a quotient of energy by a change in its correlated coordinate; the matching of force and coordinate as factors in the product that is energy being executed on due physical grounds. We have been led likewise to replace mass by a broader term inertia, where a quantity is detectable in the phenomena of more general energy-storage, that stands in essential paralleHsm with the relation of mass itself to force and kinetic energy. And the dynamical scheme has been rounded out by allowing to momentum those privileges of latency and of reappearance in the literal mechanical form, that were at the outset the monopoly of energy.

5. These comments have been attached to Lagrange's equa- 2 tions because Maxwell did in fact make them the vehicle of his thought; insisting upon sufficient detail to lift the reproach of indefiniteness, but also by a right inherent in the method passing over in silence the points where invention had thus far failed. But it was demonstrated long ago that d'Alembert and Lagrange and Hamilton have provided us with interconnected lines of approach to the same goal: except as the element of choice is directed by convenience Hamilton's principle lends equal favor and support with Lagrange's equations to the attempt to summarize a comprehensive statement in terms of energy. The former however elects to generalize for all analogous transformations upon a simple theorem: That potential energy will exhaust itself as rapidly as imposed constraints allow upon producing kinetic energy.

Beside the direct intention to indicate some reasons why dynamics leans increasingly upon energy relations, and borrows from energetics some modes of attack, these later remarks have a reverse implication as well. They intimate the belief that firm hold upon the elementary content of dynamical principles and intelligent full insight into them are not superseded, nor yet to be slighted. And the meaning here is not the mere commonplace truth that the more modest range satisfies many needs; or that historically it is the tap-root that has nourished and sustained the later growth. But recurring to what lies at the foundation is further the best preparation for the critical discrimination that must be exercised at the advancing frontier, because it holds the clews of conscious intention by which all effort there has been guided, and lends effective aid in steering an undeflected course among a medley of proposals to tolerate in concepts a figurative shading of their literal acceptation, or to condone acknowledged fictions on grounds of expediency.

6. The redistribution of emphasis upon which we have been dwelling has doubtless exercised the most penetrating influence to alter the complexion of mechanics as Newton left it, and therefore we have put it first. But there has been a second movement whose modifying effect as dynamics has grown must not be neglected, and which also like the leavening with energetics has been spread over a considerable period, though our report of its outcome can be compressed into a brief space. ^ This exhibited itself in a searching and protracted discussion on the relativeness of velocity and acceleration that did its part in contributing to clearness by removing ambiguity from a group of terms and carrying through a completer analysis of their bearings. The main concern here was not so much with the baldly kinematical side of the question; since it is plain that the final truth in that sense lies very near the surface. But the endeavor was quite specially shaped by the ambition to contrive at least soundly consistent expression for all dynamical processes that shall be recognized in physics; perhaps with some reach toward an ideal of universal and ultimate validity. The entire relativeness of those motions, which furnish leading factors of importance in decisions upon working values of dynamical quantities, is now a standard item in the opening chapters of dynamics as a corollary to choice of reference elements by agreement.

The acquirement of this point of view has therefore excluded all search for truly absolute motion and canceled the unqualified significance of the phrase which dates as far back as Newton. Since it seems flatly contradictory to unshackled relativeness, an impression may be created at first hearing that here for once the older thought has been overturned and radically revised. Yet the case is not so weak as it sounds, nor do we see, when we look below the surface, that any foundations have been affected vitally. We may be comforted to observe only another striking instance where a great mind did not everywhere and straightway hit upon most felicitous terms to describe how it dealt with powerful nascent conceptions. Newton seems to call motions absolute if they dovetailed easily with the spacious frame of physical action that his discovery of gravitation was beginning to build; and himself engrossed in a swift reconnaissance through the new region, he left later invention to amend his notation. But it is chiefly the philosophical connotations of his word absolute and not its unfitness in physics that have made it the center of futile controversy. Thus the idea that the older writers really had in mind when they spoke of absolute motion was scarcely different from one that continues to hold its ground and compels us still to separate two lines of inquiry. Because beyond the settlement of kinematical equivalences that is direct and simple since it is unhampered by any physical considerations, the questions of real difficulty remain unsettled to confront us. They have had a certain elusive character by involving a complicated and tentative estimate that must balance on the largest scale and through the whole range of physics net gain against loss in simplicity. What common background, as it were, of reference-elements is decipherable upon which the interplay of forces and of energies shall stand in simplest and most consistently detailed relief?

In consequence it has not been displaced as a tenet of orthodox dynamical doctrine that standards by which to judge of the energy, momentum and force that ought to appear in its accounts will not stand on a par if adopted at random, however interchangeable they have proved in passing upon rest, velocity and acceleration by the mathematical criteria in the more indifferent domain of kinematics. Dynamics has never hesitated to stigmatize apparent forces, for example, as spurious or fictitious in relation to its general procedure, and to revise its lists of rejections on due grounds derived from advance in knowledge and in method. The definitive resolution of uncertainties that affect reasonable decision for the questions here implied is still awaited;

of necessity that objective is not attainable conclusively while the surveys in the several provinces of physics remain both fragmentary and disconnected. Though it has been claimed indeed that secure foothold was being gained through reliance upon a reference to stellar arrangement in removing excrescences that showed by the light of its corrective tests.

7. The growing practice to designate that reference as ultimate, however, has not excluded a proper admission that its lines of specification were to be improved by whatever greater precision new discovery and analysis of it reveal definitely to be progress. And it is fairly probable that majority opinion was looking entirely in that direction for fresh landmarks until other prospects were opened with vigor in recent years. These depend upon a certain increase in freedom to retain functional forms when the time-variable is added to the coordinates and included in the group of quantities that are involved in the readjustment when a change of base in the reference is undertaken. This far-reaching proposal derived its original suggestion from optical phenomena peculiar to electromagnetism and in one sense exceptional; yet since it is the crux of this situation that a decision of universal application is sought, any unreconciled indications of alternative must be reckoned with, whereby two plans for attaining the maximum simplicity that is desired become divergent. The competitive schemes of ultimate reference cannot be weighed decisively before the ramifications of both have been traced everywhere in that detail which can afford a satisfactory conclusion through their final comparison. And for that the time does not seem ripe; especially as each thus far falls short of established universal quality by seeming to leave some combinations unreduced, or abnormal to its plan. It is therefore reassuring to our logical sense to note how the practically available devices of proximate reference persist and are neutral, save in the formulation of the limits upon which their steps of increasing precision may be declared to converge. For that their own framework is by spontaneous intention approximate can be conceded without discussion.