Dynamics, the cause of changes in the

Dynamics,branch of physical science and subdivision of mechanics that is concerned withthe motion of material objects in relation to the physical factors that affectthem: force, mass, momentum, energy.Dynamicscan be subdivided into kinematics, which describes motion, without regard toits causes, in terms of position, velocity, and acceleration; and kinetics,which is concerned with the effect of forces and torques on the motion ofbodies having mass. The foundations of dynamics were laid at the end of the16th century by Galileo Galilei who, by experimenting with a smooth ballrolling down an inclined plane, derived the law of motion for falling bodies;he was also the first to recognize that force is the cause of changes in thevelocity of a body, a fact formulated by Isaac Newton in the 17th century inhis second law of motion.  Galileo Galileo Galilei(1564–1642), sought these causative descriptions of motion: he was the first ofthe modern dynamicists.

The Italian was well–read in the workings of both theParisian and Mertonian schools. From these, he set out into the still poorlyunderstood field of kinetics. To move forward, he examined the most successfulof the ancient sciences: Archimedes’ hydrostatics. He took those principles asinspiration to examine the motion of a falling object. He utilised nomixed–body theory of matter. Instead, he treated bodies, and the media throughwhich they travel, in terms of their densities11 . Archimedes’ propositionsexplain the forces of buoyancy in equilibrium: they detail where an object willrest in a body of water.

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Galileo extended these principles from static intodynamic concepts. Archimedes explained the behaviour of bodies and theirnatural positions of rest. Galileo took this notion and applied it to bodies inmotion. His monumental postulation was that buoyancy, in addition todetermining a body’s position of rest, furthermore determines how fast a bodywill reach that position of rest. He used this force of buoyancy to try toexplain why objects fell at the speed they did. It is wrong to say that hedevised a dynamical law based on static principles. His theorems are ageneralisation of Archimedes’ static principles, which are then derivable fromGalileo’s: the converse is untrue.

These notions were not wholly new. Insteadof using a ratio of weight to resistance in order to explain motion, Galileodescribed it as a natural motion from which was subtracted the effect of themedium. Instead of having velocity determined by the ratio of a body’s weightto the medium’s resistance, it was to be determined by a natural value minussome part due to the resistance of the medium. The approach, ingenious thoughit was, led to no hoped for grand principle. The comparison between Galileo andAvempace is commonly drawn, as Avempace had postulated the same kind of thing:discarding the Aristotelian ratio.

Galileo was certainly aware of Avempace’swork, through what Averr¨oes wrote of it12. It is unfair to say that Avempacewas the originator of this sort of analysis, as it predates him by hundreds ofyears. This theory again goes back to John Philoponus, who was also well knownto Galileo.

Additionally, Avempace did not postulate Galileo’s explanations forthe causes of motion. A note should be made on his supposed discovery of theequality of fall–rates. Galileo did not make this discovery. The story is thatin 1589 he dropped various cannon balls from the Leaning Tower of Pisa, andthus the world came to know that all objects fall at the same rate.

The storyis wrong on several counts. Firstly, this experiment does not even demonstrateequal fall rates: it only shows that objects of the same composition fall atthe same rate, independent of their weight. Secondly, in 1589, Galileo did notbelieve in the equality of fall–rates.

Thatbodies of the same density fell at the same rate had been stated already byBradwardine two hundred years before. Thirdly, the selfsame experiment hadalready been performed by Giambattista Benedetti years before, and his work wasknown to Galileo. It is difficult to ascertain when Galileo concluded that allobjects fall at the same rate. He withheld publication on this subject for manyyears. The initial cause of his withholding was his own desire to bring thesubject to a completion before revealing it.

The later cause was therestrictions placed on him by the Inquisition. He knew of it by 1604, as herevealed it in correspondence with a confidant. He felt betrayed when a friendof his mistakenly revealed it to the world in the early 1630s, and onlypublished anything on the matter towards the end of his life.

SirIsaac Newton Sir Isaac Newton (1643–1728),made contributions to virtually every area of natural philosophy, mathematics,optics and astronomy. His monumental publication, Philosophæ NaturalisPrincipia Mathematica, usually called The Principia in short, was published in1687. It is likely the most influential book in the field of classicalmechanics, yet is little read.

Its purpose was set forth in its preface:…mechanics will be the science of motion resulting from any forceswhatsoever, and of the forces required to produce any motion… Newton set outto explain phenomena throughout the universe.

What lay within was to apply everywhere,and to every process. The trajectory taken by a cannon ball was to be governedby the same laws which governed the orbits of the planets. As the start of hiswork, he states his definitions of mass, momentum, inertia and forces, boththrough contact and at a distance. He then states his laws: First Law Everybody perseveres in its state of rest, or of uniform motion in a right line,unless it be compelled to change that state by forces impressed upon itthereon.SecondLaw The alteration of motion is proportional to the motive force impressed; andis made in the direction of the right line in which that force is impressed.Third Law To every action there is always opposed an equal reaction: or themutual actions of two bodies upon each other are always equal, and directed tocontrary parts.

It is broadly divided into three books, each of which alonewould eclipse almost any other. Books One and Two are titled Of the Motion ofBodies, being split into two exhaustive analyses. The third is titled The Systemof the World. The first book analyses motions in a void.

From his laws, heanalyses a multitude of motions, such as elliptic, parabolic and hyperbolicorbits around some focus. He investigates the forces that maintain these, i.e.the centripetal forces. Universal gravitation is introduced. After showing howpoint masses behave in the void under gravitation, he demonstrates that finitebodies can be treated as such. Kepler’s Laws follow directly.

The first bookorganised and systematised principles, some of which were at least dimlyunderstood before, but these principles had never been organised together intoa system of analysis for application everywhere. The second book sets out toexplain motion on Earth, where motion does not occur in a void: he sought thedetails of motion in resisting media. It is here that Newton departs from hisprogram of deducing physical behaviour based on his laws: he finds but littleuse for them. For example, in all his treatments of fluidic motion he finds noroom to apply his principle of momentum. In contrast, he conjures ingenioushypotheses to explain a myriad of things ranging from projectile motions to thespeed of sound in air. This book is a testament to Newton’s towering stature asa mathematician and dynamicist. The second book of the Principia is almostentirely new. The scholium of the first section of it reads: But, yet, that theresistance of bodies is in the ratio of the velocity, is more a mathematicalhypothesis than a physical one.

This sentiment is applicable to much of thehypotheses in the book. Today it is mostly forgotten. The book is dominated byhypothesis after hypothesis, with Newton displaying his flair for creativesolutions: often precise, often an excellent approximation, but also oftenwrong and today of only historical value. There are veins of gold hiddenwithin.

His observation that fluidic resistance is proportional to the squareof velocity can be found, as can the first description of internal fluidicfriction: The resistance arising from the want of lubricity in the parts of afluid is, cæteris paribus, proportional to the velocity with which the parts ofthe fluid are separated from each other. That most of the results wereincorrect cannot be a criticism of Newton either as a physicist ormathematician. The contribution of this book is immeasurable. For instance, itconstitutes the beginning of fluid dynamics, and studied many of its problemsfor the first time. From his efforts, his contemporaries and successors weregifted with a bridgehead from which to attack these subjects in earnest. Amyriad of potential motions through fluids is contemplated. The book is thestaging point for hydrodynamics. Newton contemplated which hullform might passthrough the water with least resistance, introducing an optimisation problemthat found application throughout the 19th century.

The third book set forthhis solutions to problems in celestial dynamics, with great success. Kepler’sLaws of Planetary Motion had resulted from Newton’s own, and he performedexhaustive analyses of the Solar System. The deficiencies in the Principia arelittle discussed. To the modern scholar, it is often impenetrable andconfusing; the language of mathematics having evolved so much since then. Acommon remark made about the Principia is that Newton strangely resorts togeometrical methods instead of his own calculus.Newtondoes not use his notation of fluxions, but even as soon as we arrive at LemmaII of Book I, the notion of calculus is present, if in an unfamiliar form. Forrigid body mechanics, there is no treatment of rotation.

Although Newton saysthat the spinning top “does not stop spinning except insofar as it is slowed byair.” there is no justification given. His statement appears directly after hisstatement of the First Law, but this law cannot tell us anything of thespinning top. Newton might have perceived that the top continues to spin, justas it would continue in linear motion if so impelled, but it is not possible toexplain the spinning top using what is within the Principia.

There is certainlyno treatment of angular momentum. The motion of a rigid body cannot bedescribed by the methods given in the Principia. There is no treatment offlexible bodies, such as the catenary curve or the vibrating string, nor isthere any analysis of the finite body pendulum. No equations of motion appearfor systems of more than two free masses, or one constrained.

A prime exampleof the field’s infancy is the three–body problem. Newton attempted to solvethis problem, but the contents of the Principia are insufficient to do so. Hedevised insightful approximations and valid inequalities, but the three– bodyproblem was insoluble from his principles. His talent in this area is evident,as his work would not be surpassed until the mid–18th century by the efforts ofEuler and Lagrange. That Newton did not solve all of mechanics’ problems is nota criticism at all, but only part of a clear– headed appraisal of what he diddo.

His achievements were monumental. He ought not to be credited with thecompletion of classical mechanics, but rather its beginnings. In the centuryfollowing The Principia’s publication, the field of mechanics swelledimmensely. For all the credit given to Newton, the world ought to be equallygrateful to his contemporaries and successors, especially Leonhard Euler, theBernoullis Jakob and John, and Joseph Lagrange. These are the men whosynthesised what we now apply today.                Workedby : Emela Hasaj


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