|Derivation of the laws of motion and
equilibrium from a metaphysical principle
|Translated by WillowW and released into the public domain. Suggestions for improving the translation are always welcome. :)|
On the 15th of April 1744, I described the principle upon which the following work is based, in the public assembly of the Royal Academy of Sciences of Paris, as reported in the Acts of that academy.
At the end of the same year, Professor Euler published his excellent book Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In a supplement to his book, this illustrious geometer showed that, in the trajectory of a particle acted on by a central force, the velocity multiplied by the line element of the trajectory is minimized.
This observation gave me great pleasure, as a beautiful application of my principle to the motion of the planets, which is determined by this principle.
From the same principle, I will now try to derive higher and more important truths.
Whether we stay locked up in our own thoughts, or venture out to survey the marvels of the universe, we find so many arguments for the existence of an all-powerful and all-wise Being, that we don't need to increase their number; rather, we should distill them down to a few solid proofs. We should at least examine the strengths and weaknesses of each argument, to give each one its proper weight; for one can do no greater damage to Truth than to base it on false reasoning.
I will not consider here arguments based on an infinite Being, arguments that seem to prove that an infinitely perfect Being must exist; for that idea is so sublime that, from our finite vantage point, we cannot find a reliable foundation upon which to build reasoned arguments.
I also will not consider the argument that the unanimity of all human beings that God exists is itself a proof of God's existence, a proof that seemed strong to that Philosopher of ancient Rome (Cicero, Tuscul., I. 13). Aside from the question of whether all human beings do indeed believe in God's existence, this argument would be refuted by only a handful of people who thought differently from all other inhabitants of the Earth. Even those who believe in Him have widely different conceptions of God, which prevents us from exploiting the general agreement about God's existence as an argument.
Finally, I will not consider the argument for God's existence from the intelligence that we have within, from the sparks of wisdom and power observed in finite beings, sparks that presumably originate from a source immense and eternal.
All these arguments seem strong to me, but they are not the type I wish to consider here.
Of those who have applied themselves to studying the Universe, several have found traces of the wisdom and power of Him who governs it. The more the field of physics has progressed, the more arguments we find for His existence. However, some have been confused by the character of the Divinity revealed in Nature, while others, almost religiously zealous, have given too much weight to some arguments, considering them proofs when they were not.
Perhaps the rigor of logic may be sometimes relaxed to reach a useful but uncertain conclusion, when it lacks sufficient confirmation. However, the arguments for God's existence are sufficiently strong and sufficiently many that we can assess them rigorously to determine which arguments are the most certain.
I will not pause to consider the arguments for a supreme Being that the ancients derived from the beauty, order and arrangement of the universe, such as those related by Cicero (Tuscul. I. 28, 29) and Aristotle (De Nat. Deor. II. 37, 38). They knew too little of Nature to be in a position to admire it. Rather, I consider the arguments of a philosopher who made such grand discoveries that he could well judge the truly marvellous in Nature, and whose reasoning was more precise than that of the others.
Newton was moved most of all by arguments derived from the study of the universe, although his deep-thinking mind derived other arguments as well.
This great man believed (Opticks III. Book. Query 31) that the motion of celestial bodies demonstrated the existence of Him that governs them. Six planets — Mercury, Venus, Earth, Mars, Jupiter and Saturn — revolve about the Sun. All the planets rotate in the same direction in orbits that are more-or-less concentric. (Nevertheless, there are other bodies, the comets, that follow totally different orbits, moving in all directions and in every region of Heaven.) Newton believed that such uniformity of motion could only result from the will of a supreme Being.
Newton also found strong arguments in less exalted objects. For him, the obvious similarity in the construction of animals, their organization and suitability for their purpose, were convincing arguments for the existence of an all-powerful and all-wise Creator (Theol Astron. de Derham, Theol. Phys. du meme, Theol. des Insectes de Lesser).
A host of physicists following Newton have found arguments for God in astronomical bodies, in insects, in plants and even in water (Theol. de l'Eau de Fabricius).
Let us not disguise the weakness of these arguments; indeed, to better understand the abuse which some have made of proofs of God's existence, let us critique even arguments as strong as those of Newton.
Newton states that the uniform motion of the planets reveals an Intelligent Designer, that it is not possible that a soulless universe could make them rotate in the same direction and in concentric orbits.
Newton could also have added that all of these planets move in the same plane. All of their orbits are contained within a solid angle that is the 17th part of a sphere. Taking the plane of the Earth's orbit as a reference, the random probability that five other orbits would fall within the same solid angle is , i.e., 1419856 to 1 against.
Thus, thinking as Newton did (i.e., that all celestial bodies are attracted to the sun and move through empty space), it is extremely improbable that the six planets would move as they do. However, the probability is not zero and, hence, the uniformity of planetary motion is not a necessary proof of an Intelligent Designer.
There is another consideration. The two alternatives, Intelligent Design versus pure chance, are based on our inability to find a physical cause for the uniformity of planetary motion within Newton's system. However, other philosophers have hypothesized a fluid that transports the planets or at least regulates their motion; if true, that might explain the uniformity of planetary motion (rather than an Intelligent Designer or pure chance) and would be no more proof of God's existence than any other motion imposed on matter.
I don't know whether Newton's argument based on the construction of animals is much better. A proof based on the similarity of many animals would seem to be refuted by the infinite variety observed in others. Without comparing their most basic parts, how shall we assess the similarity of an eagle with a fly, a stag with a slug, a whale with an oyster? Other philosophers have argued for God's existence from the variety of forms and I don't know which argument has a better basis.
The argument based on the suitability of the different animal parts for their needs seems more solid. Feet seem to be made for walking, wings for flying, eyes for seeing, mouths for eating and other parts for reproducing their kin. All these things seem to feature intelligent design in their construction. This argument carried as much weight with the ancients as it did with Newton. In vain, the greatest opponent of Providence (Lucretius, Book IV) argued that the usage of these organs was not by design but a consequence of their construction, that the eyes, ears, tongue, etc. formed by chance and that animals merely exploited them to see, to hear, to speak, etc.
But perhaps it is not miraculous that suitable organs are found in species that actually exist, since they help those animals to survive. One might argue that Chance has produced a countless number of individual animals, of which a few were constructed so that they could meet their own needs; the vast number of other individual animals perished since their parts were not suitable for survival. Animals without mouths cannot live; those lacking organs of generation cannot reproduce themselves. The few animals that have survived are those with well-ordered parts suited to survival; thus, the animals that we see today are but the smallest part of those produced by a soulless universe.
Almost all of the modern authors that consider physics and natural history merely extend the old arguments of the organization of animals and plants, pushing them into ever finer details of Nature. To avoid the vulgar, gory details, I mention only one author who argues for God from the folds in the skin of the rhinoceros. The skin of that animal is so thick and hard that, without such folds, it would be unable to move. Doesn't it demean the great truth of God's existence, if it must be proven by such arguments? And what shall we say to the author who argues against Providence from the fact that the tortoise's shell has neither joints nor folds? The reasoning seems the same as the rhinoceros-skin argument, with the same weight. Let us leave such trifling arguments to those who haven't learned to laugh at them yet.
Another type of philosopher goes to the other extreme. Finding all too few traces of intelligent design in Nature, he disregards all final causes and believes that world formed itself as it is from pure matter and its motion. The first type sees an Intelligent Designer everywhere; the other sees Him nowhere, believing rather that a soulless mechanics was able to form bodies as sophisticated as animals and plants and to bring about all the wonders of the universe (Descartes, Princip. L'Homme de Descartes).
It is interesting to note that Newton was not impressed by Descartes' great argument for God's existence derived from the idea of a perfect Being, nor by other metaphysical arguments that we have mentioned; yet Newton's own arguments for God's existence from the uniformity and suitability of different parts of the universe would not have seemed like proofs to Descartes.
It has to be admitted that these "proofs" have been abused; some give the arguments more weight than they deserve, whereas others multiply them too much. The bodies of animals and plants are machines too sophisticated for our understanding; their smallest parts escape our notice and we are too ignorant of their usage and purpose that we can presume to judge how much wisdom and power went into their construction. Some of these machines seem highly perfected, whereas others seem like coarse sketches. Many parts seem useless or even harmful based on our present knowledge, unless we assume in advance that they were sent into the world by a all-wise and all-powerful Being.
Finding traces of order and suitability in the construction of animals may lead us to unsettling conclusions. The snake, which neither walks nor flies, could not evade the pursuit of other animals if it had not been blessed with such flexibility that it slithers faster than many animals can walk. It would die of cold in winter, except that its long and pointed form suits it to tunnel underground; it would be hurt by its continual slithering or passing into the hole where it hides, were it not for its smooth and scaly skin. Is not the snake something wonderful? Yet these wonders only help to preserve an animal that kills human beings. "Oh!" one replies, "you don't know the utility of snakes. They appear to be necessary in the universe; they must contain some excellent remedy of which we are presently unaware. Therefore, let us be silent or, at least, not be surprised by the beautiful mechanisms in animals that seem harmful."
The writings of naturalists are full of such reasonings. Follow the development of a fly or an ant, and you will find the wonders of Providence in the insect's eggs, in how it feeds its children, in how it seals itself in a chrysalis and re-emerges after metamorphosis. All these wonders serve to produce an insect that bothers human beings, an insect that will be eaten by a bird, or get caught in a spider's net.
Even though some people may find proofs of God's existence in these examples, others may be reinforced in their scepticism.
Even the greatest of spirits, respected for their piety as well as their enlightenment (P. Malebranche, Medit. Chret. & Metaph., Medit. VII), cannot deny that the suitability and order observed in the universe is not so perfect as to be a proof of an all-wise and all-powerful Being. Evils of every type, the disorder, the vices, the grief: these are hard to reconcile with an world governed by an all-wise and all-powerful Ruler.
One might argue, "Consider the Earth, covered mostly by water or by sharp rocks, icy regions and burning sands. Consider the habits of those who live there, how they lie, steal, and murder; vice is more common than virtue everywhere. Many of those miserable people are despairing, troubled by gout, stones and other diseases that make their life unbearable; almost everyone is vexed by sorrow and disappointments."
Several philosophers seem to have had this perspective and, forgetting all the beauties in the universe, sought to justify God for having created such imperfections. In particular, to shield God's wisdom from criticism, some philosophers seem to diminish His power, saying that God made the world as good as it could be made (Leibnitz, Theod. II. part. N. 224,225), i.e., that of all possible worlds, this one is the best, despite its faults. Other philosophers preserve His power at the expense of His wisdom, saying that God could have made a better world than this, but it would've required a more complicated mechanism and that He had the mechanism more in mind than the perfection of the work (Malebranche, Medit. Chret. & Metaph. VII). They give the example of a painter who thinks that a circle drawn by hand shows his skill more than one drawn with a compass, never considering that instruments can help in making figures more complex and regular.
These arguments are not satisfying, but neither can they be soundly refuted. Still, a true philosopher should neither be dazzled by the order and suitability of the parts of the universe, nor seek to grasp things that lie beyond Nature. Despite the disorder observed in Nature, one finds enough traces of the wisdom and power of its Author that one cannot fail to recognize Him.
I will not discuss another type of philosopher who refuses to recognize any evil in Nature: Everything that is, is good. (Pope, Essai sur l'homme).
This proposition is untenable unless we assume beforehand the existence of an all-powerful and all-wise Being. If we derive this proposition from that assumption, it is a simple act of faith. Although it seems to honor the supreme Intelligence, it also seems like just submission to necessity. It is more a consolation for our sorrows, than praise for our happiness.
Let me return to the arguments based on the study of Nature.
Those who have reviewed such arguments have not examined their strength nor scope. Many things in the universe suggest that it is governed by a blind power. On all sides, we see consequences of effects leading to some destination; but this does not prove intelligent design. We must rather seek signs of God's wisdom in the goals of His designs; wonderful skill in execution cannot compensate for a senseless undertaking. Skill applied senselessly is not admirable; one would blame the builder all the more if he used skill to construct a machine that was useless or even dangerous.
How does it help to wonder at the regularity of the planets, how they rotate in the same direction, in almost the same plane, and in orbits that are almost geometrically similar, if we cannot see why it is better that they move thus and not otherwise? All those poisonous plants and deadly animals are produced and preserved so carefully by Nature -- how can we recognize from them the wisdom and kindness of Him who created them? If we found only such things in Nature, it might well be the work of demons.
Truly our perspective is limited to where we are; we cannot see far enough to appreciate the order and interconnectedness of things. If we could, we would undoubtedly find the marks of God's wisdom as well as His intelligence in its execution. But, given our limitations, let us not confuse the two attributes. For although an infinite intelligence necessarily brings with it wisdom, a finite intelligence may yet lack wisdom; and there is as much evidence showing that the universe is a soulless machine, as showing it to be the work of an Intelligent Designer.
We should not seek the supreme Being in little details, in the parts of the universe of whose relationships we know too little; rather, we should seek Him in universal phenomena that allow no exception and whose simplicity is entirely exposed to our view.
Admittedly, such research is more difficult than studying an insect, a flower, or similar things that Nature offers to our eyes every moment. But we can be aided in our difficult march by a guide, one who is sure-footed although he has not yet gone where we would like to go.
Until now, the goals of mathematics have rarely gone beyond coarse needs of the body and useless speculations of the spirit. The truths that it has uncovered are mostly those relating to dimensions and numbers. One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simiplicity of the objects considered by mathematics.
Let us see whether we can find a better use for mathematics. Mathematical arguments for God's existence would have the obvious certainty characteristic of geometrical truths. Those who doubt metaphysical reasoning would believe a mathematical argument more readily, whereas those exposed to the usual arguments would find mathematical arguments more elevating and precise.
Therefore, let us not stop at simple speculation about the marvels of Nature. The organization of animals, the precise and many parts of insects, the enormous sizes and distances and revolutions of astronomical bodies, such arguments are better suited to amaze our mind than to clarify it. The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws.
I could have started from mathematical laws that are confirmed by experience, and tried to show how they reflect the wisdom and power of the supreme Being. But those laws can be uncertain since they are often not based on rigorous proofs or purely geometrical hypotheses. Therefore, I considered it more certain and useful to deduce mathematical laws from the attributes of an all-powerful and all-wise Being. If those derived laws are actually found in the universe, then the proof that this Being exists (and created those laws) is more certain.
But, one might object, what if the laws of motion and equilibrium are a necessary consequence of the nature of matter, and contain no arbitrariness? Wouldn't it be wrong to ascribe to Providence something that is absolutely necessary?
If it is true that the laws of motion and equilibrium are indeed absolutely necessary consequences of the nature of matter, that proves all the more the perfection of the supreme Being. Everything is so arranged that the blind logic of mathematics executes the will of the most enlightened and free Mind.
Certain philosophers of antiquity held that movement does not exist. An overly subtle use of their minds denied the evidence of their own senses. The difficulties they had in conceiving how a body could move, made them deny that bodies moved or even could move. Without delving into the arguments upon which they tried to base their opinion, we note that one cannot deny motion except by arguments that deny the reality of objects outside ourselves, arguments that reduce the universe to us, and all phenomena to our perceptions.
It is true that we know motion only through our senses; but how many things do we know otherwise? The motive force (i.e., the ability of a moving body to move other bodies) is just terminology filling in for understanding and has no meaning beyond the results of phenomena. It is only mental habit that prevents us from realizing how miraculous it is that motion can be passed from one body to another. Once our eyes have opened, nothing is so striking. For those who have never thought about it, it doesn't sem mysterious; by contrast, those who have meditated on it may despair of ever understanding it.
Imagine someone who had never touched a body and had never seen them collide, but who had some experience with mixing colors, and let this person see a blue body moving towards a yellow body. If asked about what would happen when the two bodies collided, he might say that the blue body would turn green when it hit the yellow body. But would he guess that the two bodies would join together and move at a common speed? or that one body would transfer part of its motion to the other so that it moves in the same direction at a different speed? or that it would be reflected in the opposite direction? I don't think it would be possible to guess these things.
We have experienced the impenetrability of a material body, we have proven to ourselves that a certain force is required to change its motion, we have seen that the collision of two bodies causes them to stop or be reflected or displace a resting body or, more generally, change their speed. Yet how do such changes occur? What is this power that bodies seem to have to act on one another?
We observe that some particles of matter are in motion, while others are at rest. Hence, motion is not an essential property of matter; rather, it is a state in which matter may find itself — or not. We see only that material bodies can extract motion from each other.
The material particles moving in Nature received their motion from an external cause that is unknown hitherto. The particles are themselves indifferent to motion or rest; if at rest, they stay so, whereas moving bodies continue to move until something changes their motion. When a moving particle of matter encounters another particle at rest, it passes on some or all of its motion. Every collision of two bodies (whether one is at rest and the other moves, or both are moving) is accompanied by a change in motion. The impact causes this change; it would be absurd to say that a particle moves another particle, given that it can't even move itself.
To identify the original cause of motion, the greatest philosopher of antiquity resorted to a Prime Mover, itself unmovable and indivisible. A modern philosopher has not only recognized God as the author of all motion of material particles, but also argued that God's direct intervention is continually necessary for every change and redistribution of motion. Not being able to understand how bodies can move other bodies, he denied that they did and concluded that when bodies hit or press on other bodies, it is God who makes them move; their impact is nothing but a prompting for God to move them.
These philosophers assign the cause of motion to God because they don't know where else to put it; they resort to an immaterial Being, since they can't imagine that matter should have the power to produce, distribute and destroy motion. But it's important to realize that all the laws of motion and equilibrium should be based on a suitable principle, if we are to use them to prove God's existence, regardless of whether bodies can act directly on each other, or whether they use some other interaction as yet poorly understood.
The simplest law of Nature is that of equilibrium, which has been known for many centuries; but until now, it has seemed to have no connection to the laws of motion, which were much more difficult to discover.
Research into motion was not to the liking (or perhaps not within the scope) of the ancients, so that we may consider it as a completely new science. How could the ancients have discovered the laws of moiton, given that some philosophers reduced all their speculations about motion to sophistic disputes,whereas others denied that motion existed at all?
More industrious and sensible philosophers judged only that it would be difficult to understand the principles of motion, and invented reasons to despair of ever knowing them, as an excuse to get out of doing experiments.
A true philosopher does not engage in vain disputes about the nature of motion; rather, he wishes to know the laws by which it is distributed, conserved or destroyed, knowing that such laws are the basis for all natural philosophy.
The great Descartes, boldest of the philosophers, sought such laws, but was mistaken. However, time brought the subject of mechanics to a kind of maturity and those laws, unknown for centuries, were developed suddenly in several places at once. Huygens, Wallis and Wren discovered them at the same time. Many mathematicians after them have confirmed these laws, although they arrived at these laws by different routes.
Nevertheless, although mathematicians agree today on the more complicated case, they do not agree on the simpler case. Everyone agrees about the distribution of motion in the collision of perfectly elastic bodies, but not in the collision of perfectly inelastic bodies; some even believe that the distribution of motion cannot be calculated in the latter case. This embarassing failure has caused some to deny the existence and even the possibility of perfectly inelastic bodies, claiming that bodies believed to perfectly inelastic are actually elastic bodies whose hardness prevents us from seeing the deformation and recovery of its shape.
These people point to experiments on bodies commonly called "hard" and "undeformable" showing that these bodies are actually deformable and elastic. For example, when spheres made of ivory, or maple wood, or glass collide, they seem to have deformed during the collision, only to recover their shape immediately afterwards. This may be seen by painting one of the spheres with a color that comes off and stains the other sphere; the size of the stain after the collision shows that the spheres were flattened during the collision, although no trace of the deformation remains afterwards.
These people then add metaphysical arguments to these arguments. They require that every process in Nature take a certain time, and obey a law of continuity. They object that when a perfectly inelastic body collides with an impassable barrier, its speed must go to zero instantly, without passing through any intermediate values; or its speed instantly is transformed into its opposite, without passing through zero.
But I confess, I don't feel the force of this argument. I don't think we know enough of the means by which motion can be produced or destroyed to say whether it must follow the law of continuity. If you assume that speed must change by degrees, doesn't it change discontinuously between degrees? Even imperceptible discontinuous changes would violate the law of continuity as much as the sudden destruction of the universe.
The experiments suggest that it is easy to confuse hardness with elasticity, but they do not show that all hard bodies are elastic. On the contrary, those who have considered the impenetrability of bodies suggest that perfect inelasticity is a necessary consequence of matter. Most bodies are deformable because they are made up of particles that can move relative to each other when stress is applied; however, the fundamental particles, the elements that make up all other bodies, these must be bodies that are perfectly inelastic, undeformable and unchangeable.
The more one considers elasticity, the more it seems to require that the elastic body have parts that can move relative to one another under stress.
Hence, it seems more reasonable to assert that the fundamental particles of matter are perfectly hard, rather than to pretend that perfectly hard bodies do not exist in Nature. Nevertheless, I do not know whether our present knowledge allows us to make either assertion definitely. However, one of the main reasons for denying that perfectly hard bodies exist was that we were unable to derive the laws of motion transfer for hard bodies.
Descartes agreed that perfectly hard bodies exist, and believed to have discovered the laws of their motion. It started from a principle that seems reasonable, namely, that total momentum is conserved in Nature. Unfortunately, he derived false laws, because his initial principle, the conservation of total momentum, is not true.
The philosophers who followed Descartes noted another conservation law: that of kinetic energy, which is one-half times the mass times the speed squared. These philosophers did not derive their laws of motion from this conservation law, but rather the reverse; they derived the conservation law from their laws of motion. However, since kinetic energy is conserved only for perfectly elastic bodies, that reinforced them in their belief that only elastic bodies exist in Nature.
Momentum is conserved only in special cases, and kinetic energy is conserved only for certain types of bodies. Neither one represents a universal principle, a general law of motion.
If one studies these principles and the reasoning their authors used, it is surprising that they ever were able to derive them; one suspects that they worked backwards from experience, rather than deducing them from basic principles. Those who argue most rigorously recognize that the principle used to derive the collision laws for elastic bodies would not be applicable to perfectly inelastic bodies. Moreover, the laws of mechanical equilibrium are independent of the conservation laws used to derive laws for collisions of elastic and inelastic bodies.
After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based, a principle that covers both elastic and inelastic collisions and describes the motion and equilibrium of all material bodies.
This is the principle of least action, a principle so wise and so worthy of the supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits. In the collision of bodies, motion is distributed such that the quantity of action is as small as possible, given that the collision occurs. At equilibrium, the bodies are arranged such that, if they were to undergo a small movement, the quantity of action would be smallest.
The laws of motion and equilibrium derived from this principle are exactly those observed in Nature. We may admire the applications of this principle in all phenomena: the movement of animals, the growth of plants, the revolutions of the planets, all are consequences of this principle. The spectacle of the universe seems all the more grand and beautiful and worthy of its Author, when one considers that it is all derived from a small number of laws laid down most wisely. Only thus can we gain a fitting idea of the power and wisdom of the supreme Being, not from some small part of creation for which we know neither the construction, usage nor its relationship to other parts. What satisfaction for the human spirit in contemplating these laws of motion and equilibrium for all bodies in the universe, and in finding within them proof of the existence of Him who governs the universe!
Whether at rest or moving, all material bodies have a certain force that works to maintain them in their current state of motion; this force, denoted inertia, is proportional to the amount of matter they contain.
The two properties, impenetrability and inertia, of material bodies are always opposed to one another in Nature, making laws necessary to bring them into harmony. When two bodies encounter one another, they cannot penetrate one another and, therefore, the rest of one and the motion of the other (or the motion of both) must be changed. Since this change depends on the force with which the two bodies collide, let us examine the nature of the collision and the factors affecting its force; and if we cannot obtain a sufficiently clear idea of its force, at least let us determine the conditions under which the force is the same.
We make several assumptions here that are common to all researchers who have studied the laws of motion. Specifically, we assume that the bodies collide directly, i.e., that their centers of gravity move only along in the straight line that connects them; that the point of contact in the collision lies on this line; and that the tangent surfaces at the point of contact are penpendicular to this line. This last condition is always true if the two bodies are solid spheres composed of a homogenous material, as we assume here.
If a body moves with a certain speed and encounters a second body at rest, the impact would be the same as if the second body, moving with the speed of the first, had collided with the first body at rest.
If two bodies move in opposite directions and collide, the impact would be the same as if one body were at rest and the second body moved with a speed equal to the sum of the original two speeds.
If two bodies move in the same direction and collide, the impact would be the same as if one body were at rest and the second body moved with a speed equal to the difference of the original two speeds.
Thus, whenever two bodies collide, the impact is generally the same as long as their relative speed (i.e., the sum or difference of their speeds) is the same, regardless of their individual speeds. The magnitude of the impact depends only on the relative speed of the two colliding bodies.
The truth of this proposition is easy to see, by imaging both bodies being transported on an invisible, massless plane moving such that the speed of one body vanishes and the second body has either the sum or the difference of the original two speeds. The impact of the two bodies on the moving place would be the same as on a stationary plane.
Let us now consider the effects of elasticity on the collision.
In a perfectly inelastic body, the parts are inseparable and inflexible; hence, its shape simply cannot be changed.
In a perfectly elastic body, the parts may deform, but always recover to exactly their original shape and situation. We do not undertake to explain the origin of this elasticity, merely its effects.
I will not discuss "squishy" or fluid bodies at all, since they are composed of inelastic or elastic bodies.
When two inelastic bodies collide, their parts are inseparable and inflexible; hence, the impact can affect only their speed. The two bodies press and push until the speed of the first equals the speed of the second. After the impact, inelastic bodies move with a common speed.
However, when two elastic bodies collide, while the are pressing and pushing, the impact serves to deform their parts. The two bodies do not remain touching each other; the restoration of their former shape causes them to rebound from one another, separating from one another at the same speed with which they approached. For their relative speed was the sole cause of their initial collision and the rebounding should produce an effect equal to that which caused the collision; thus, the final relative speed of the two bodies should equal the initial relative speed, albeit with opposite direction. Hence, the final relative speed of two elastic bodies after the collision should be the same as the initial relative speed.
Let us now seek the laws that govern the distribution of motion among colliding bodies, whether they be elastic or inelastic.
We will derive these laws from only one principle and, from the same principle, we will derive the laws of mechanical equilibrium.
When a change occurs in Nature, the quantity of action necessary for that change is as small as possible.
The quantity of action is the product of the mass of the bodies times their speed and the distance they travel. When a body is transported from one place to another, the action is proportional to the mass of the body, to its speed and to the distance over which it is transported.
Let there be two inelastic bodies of masses m1 and m2 moving in the same direction with speeds v1 and v2, respectively. Let the first mass move more quickly, so that it overtakes the second mass and collides with it. After the collision, let the common velocity of the two bodies be vf such that v2 < vf < v1. The change in the universe is that, whereas the first mass was moving at a speed v1 and was covering a distance v1 per unit time, it now moves only at a speed vf and covers only a distance vf per unit time; and whereas the second mass was moving only at a speed v2 and was covering only a distance v2 per unit time, it now moves at speed vf and covers a distance vf per unit time.
This change would be the same if
The motion of these immaterial planes conveying the masses m1 and m2 are the same, regardless of whether the masses are moving relative to these planes or are at rest. Hence, the quantities of action produced in Nature are and , the sum of which should be minimized. Thus, we have
from which one can derive the final speed
In this case where the two bodies are moving in the same direction, the quantity of momentum produced and destroyed is the same; the total momentum is constant, being the same after the impact as beforehand.
It is easy to extend the same reasoning to the case where the two bodies are moving towards each other, by making the second speed v2 negative. In that case, the final speed is
If the second body is at rest before the impact, v2 = 0, and the final speed is
If the first body encounters an impassable barrier, one can consider that barrier as a body of infinite mass at rest; since m2 is infinite, the final speed vf = 0.
Now let us consider what happens to elastic bodies, by which I mean perfectly elastic bodies.
Let there be two elastic bodies of masses m1 and m2 moving in the same direction with speeds v1 and v2, respectively. Let the first mass move more quickly, so that it overtakes the second mass and collides with it. Let u1 and u2 represent the speeds of the two bodies after the collision; the sum or difference of these speeds is the same as that before the collision.
The change in the universe is that, whereas the first mass was moving at speed v1 and was covering a distance v1 per unit time, now it moves at speed u1 and covers a distance u1 per unit time; and whereas the second mass was moving at speed v2 and was covering a distance v2 per unit time, now it moves at speed u2 and covers a distance u2 per unit time.
This change would be the same if
The motion of these immaterial planes conveying the masses m1 and m2 are the same, regardless of whether the masses are moving relative to these planes or are at rest. Hence, the quantities of action produced in Nature are and , the sum of which should be minimized. Thus, we have
For elastic bodies, the relative speed after the impact should equal the relative speed before the impact; hence, we have u2 − u1 = v1 − v2 or, rather, u2 = u1 + v1 − v2 and, thus, du2 = du1. Substitution into the preceding equation yields the final speeds
It is easy to extend the same reasoning to the case where the two bodies are moving towards each other, by making the second speed v2 negative. In that case, the final speeds are
If the second body is at rest before the impact, v2 = 0, and the final speeds are
If the first body encounters an impassable barrier, one can consider that barrier as a body of infinite mass at rest. In that case, the final speed u1 = − v1, i.e., the first mass rebounds at the same speed with which it struck the barrier.
If one takes the sum of the kinetic energies, one sees that they are the same after the impact as before; thus,
the sum of the kinetic energies is conserved after the impact. However, this conservation applied only to elastic bodies, and not to inelastic bodies. The general principle that applies to both types of bodies is that the quantity of action required to cause a change in Nature is as small as possible.
This principle is so universal and so fruitful that one can also derive the law of mechanical equilibrium from it. At equilibrium, there is no difference between elastic and inelastic bodies.
I now suppose that two bodies are attached to a lever, and I seek the point about which they remain in equilibrium. Thus, I seek the point about which, if the lever moves slightly, the quantity of action is as small as possible.
Let L be the length of the lever (which I suppose to be massless), and let two masses m1 and m2 be placed at either end. If z represents the distance from the first mass m1 to the equilibrium point being sought, then L − z represents the corresponding distance to the second mass m2. Obviously, if the lever rotates slightly about a point, the two masses describe geometrically similar arcs, whose size is proportional to their respective distances from the point of rotation. Thus, these arcs are the distances traveled by the bodies and also represent their speeds per unit time. Hence, the quantity of action is proportional to the product of the mass of each body multiplied by the square of its arc length; or, equivalently (since the two arcs are geometrically similar), the quantity of action is proportional to the product of the mass of each body multiplied by the square of its distance to the point of rotation, i.e., m1z2 and . The sum of these two terms should be minimized, giving the equation
from which the equilibrium position may be derived
This is the basic law of mechanical equilibrium.
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