Leibniz And Dynamics: A Dialogue

by Phil Valenti

{Xena,} a young student.
{Academos,} a middle aged professional.
{A Philosopher.}

{Xena:} Greetings to you, Philosopher! I’m so glad you came along just now. Academos here is trying to convince me of his latest opinions about science, which have me awfully confused.

{Academos:} That’s right, Sir Philosopher! I’ve been reading about Leibniz, of whom you think so highly. Even you can’t deny that his idea of the “living force,” which is somehow implanted by God into matter, is nothing more than medieval metaphysical nonsense.

{Philosopher:} Well, my young friends, I can’t deny that his idea is metaphysical, but it is far from nonsense. In fact, all of modern technology is based on it.

{Xena:} But Academos gave me an example, which is hard to refute, although I’d like to.

{Philosopher:} Let’s hear this example.

{Academos:} Well, it concerns the issue of how to measure force. Descartes says that force equals the mass of an object multiplied by its velocity, whereas Leibniz constantly insists that force is proportional to the mass of an object multiplied by the SQUARE of the velocity. But this is just a phony dispute over words.

{Philosopher:} Please explain.

{Academos:} For example, it doesn’t matter whether you measure distances in miles or kilometers, as long as you’re consistent. Alexandria, Egypt is twice as far from Rome than it is from Athens, whether you measure it in miles or kilometers. You see, it depends upon what yardstick you use, and that’s just a matter of personal preference. In the same way, some people may choose to measure force by mass times velocity, and others by mass times velocity squared.

{Xena:} I still like to think that there must be some way of discovering what is true and what isn’t.

{Philosopher:} I’m happy to have this discussion with both of you, but I warn you, it will take much concentration. The rewards, however, will be great. Believe me when I say, that the results of this investigation will have the most profound impact on your entire view of the nature of the Universe and of the future of Mankind. It may even transform your conception of the value and purpose of your own life.

{Academos:} There you go, exaggerating again!

{Xena:} It sounds exciting, but I can’t see how that’s possible.

{Philosopher:} Let us start out by assuming that the force of a moving body equals its mass multiplied by its velocity. This means that a body weighing ten pounds and moving at one mile per hour, will have the same force as a body weighing one pound and moving at ten miles per hour, correct?

{Academos:} If you choose to measure force that way, that is the result you will get.

{Philosopher:} This also means that a body weighing 100 pounds and moving at 1/10th of a mile per hour, will have the same force as a body weighing 1/10th of a pound and moving at 100 miles per hour.

{Academos:} That’s right, as long as the ratios are the same.

{Philosopher:} In other words, a body weighing 1000 pounds and moving at 1/100th of a mile per hour, will have the same force as a body weighing 1/100th of a pound and moving at 1000 miles per hour?

{Academos:} Absolutely.

{Philosopher:} Are you sure? Think what would happen if you were hit by those objects.

{Xena:} Wait, I see what you mean. I probably would hardly feel a thousand-pound object moving so slowly, but I can’t imagine what a small object moving so fast would do to me. It probably would have the force of a bullet. Hey, I might be blown away by something moving that fast!

{Philosopher:} This is Leibniz’s point, when he says that the “active or living force appears in impact.” Remember, Leibniz also talks about “calculating the force through the effect produced in using itself up. For I here refer not to any effect,” he says, “but to one produced by a force which completely expends itself and may therefore be called violent.”

{Xena:} I can see how the effect would be violent for sure.

{Academos:} Let’s not jump to conclusions. There are other authorities besides Leibniz. What about Archimedes? In his study of the lever, he showed that a body weighing 10 pounds, which is one foot from a fulcrum, will balance a body weighing one pound, which is ten feet from the fulcrum. In other words, the weight and distance from the fulcrum are reciprocally proportional. Why shouldn’t the same relationship hold for mass and velocity?

{Xena:} Now I’m confused again.

{Philosopher:} Archimedes was certainly a great thinker, and this is a good point, because it will help us see the difference between Mechanics, a perfectly valid science which pertains to the ancient machines like the lever, pulley, inclined plane, wheel and screw, and Leibniz’s new science of Dynamics, which brought about the steam engine, and every technological advance after that as well!

Leibniz refers to Galileo, who, Leibniz says, “paradoxically called the IMPACT OF PERCUSSION an infinitely large force as compared to the simple tendency of gravitational force.”

Think about a pile driver. If you lay the pile driver gently on top of a post stuck in the ground, the weight of the pile driver will push the post a little further into the ground. You might conclude that the total force of the pile driver has been expended. But, if you drop the pile driver down onto the post from a distance above it, it will drive the post further into the ground. In other words, the pile driver has somehow accumulated more force by virtue of its motion. If you drop it again, it will drive the post a little further down. It has once again accumulated more force.

{Xena:} This is awesome.

{Academos:} This is ridiculous! How can motion change an object? You make it sound as if there’s something mysterious inside things which “comes alive” when bodies move, whereas an authority as great as Descartes shows that bodies are just passive things that exist in empty space.

Descartes says that “the nature of matter or of body in its universal aspect, does not consist in its being hard, or heavy, or colored, or one that affects our senses in some other way, but solely in the fact that it is a substance extended in length, breadth and depth.” In other words, “that the nature of body consists … in extension alone.” That’s why you can multiply and divide material bodies just as if they were purely geometrical or mathematical entities.

For example, if a freight car weighing 1000 pounds and moving at 10 miles per hour, collides and couples with another freight car weighing 1000 pounds which is stationary, the two of them will move in the direction of the first freight car, at a velocity of five miles per hour. This is because 1000 X 10 plus 1000 X 0 equals 10,000, which is the total mass X velocity before they couple, and also 2000 X 5 equals 10,000, which is the total mass X velocity after they couple, not counting friction. Haven’t you ever heard of the conservation of momentum?

{Xena:} I know I never heard of it.

{Philosopher:} Don’t worry, Xena. Let us keep up our discussion, and, with Leibniz’s help, you will see that you are much smarter than Descartes, despite his fame. Let us assume that the freight cars will behave approximately in the way Academos describes. The question is: why don’t both freight cars move at TEN miles per hour after they collide and couple?

{Academos:} I already did the calculation.

{Philosopher:} But I am asking WHY? Or if a small body collides with a large body at rest, why doesn’t the small body carry the large one along with it, without losing any velocity? In other words, is there anything in the concept of body as mere passive “extension,” to account for INERTIA?

{Academos:} But inertia is simply a physical law, which says that a body at rest will tend to stay at rest, unless acted upon by an outside force. Similarly, a body moving at a constant velocity will tend to continue its motion, unless an outside force acts upon it.

{Philosopher:} In other words, Academos, there is a certain “resistance” inherent in material things. It takes an effort, or work, to move something, or to change its velocity, whether it’s a freight car, or anything else. But if objects were just mathematical or purely geometrical entities, they would be purely indifferent to motion, wouldn’t they?

{Xena:} Well, I for one can’t imagine circles, squares and triangles “resisting” geometric constructions, or numbers “resisting’ being added!

{Philosopher:} Now you see the paradox!

“If the essence of a body consisted in extension,” Leibniz writes, “this extension alone should suffice to account for all the properties of the body. But that is not the case. We observe in matter a quality which some have called natural inertia, through which the body resists motion in some manner, in such wise that some force must be applied to set it into motion (not even taking into account the weight), so that it is more difficult to budge a large body than a small one. For example, if the body A in motion meets the body B at rest, it is clear that if B were indifferent to motion or rest, it would let itself be pushed by A without resisting it, and without diminishing the speed or changing the direction of A; and after the impact, A would continue its path, and B would accompany it ahead. But it is not so in nature. The larger the body B, the more it will diminish the speed of A, until A is forced to rebound from B, if B is very much larger than A….

“All of this shows that there is in matter something else than the purely Geometrical, that is, than just extension and bare change. And in considering the matter closely, we perceive that we must add to them some higher or metaphysical notion, namely, that of substance, action, and force.”

{Academos:} I’m amazed at how you insist on explaining everyday things with abstract metaphysical constructs!

{Xena:} This is exciting! I want to hear more.

{Philosopher:} Yes, we have a bit more work to accomplish before reaching our objective, which is to demonstrate why Leibniz is right, and Descartes wrong, and that the force, or power, of a body in motion is proportional to the mass and the square of the velocity.

{Academos:} If you can show how such a practical result follows from all of this metaphysical mumbo-jumbo, I will be very surprised.

{Philosopher:} You would have to rethink all of your assumptions about the world, which is a good thing. Let us begin by analyzing what happens to a heavy body in free fall.

{Xena:} What do you mean by “free fall”?

{Academos:} He just means a body falling under the influence of gravity.

{Xena:} Oh. This sounds like the case of the pile driver we discussed before.

{Philosopher:} That’s right. Take, for example, this paperweight, which I place on the ground in front of me. Now, I pick it up, and lift it about four feet above the spot where it was lying. This involved some effort, or work, on my part, which I have, so to speak, transferred to the paperweight, with the result that the paperweight has been raised four feet above the ground.

{Xena:} I follow you so far.

{Philosopher:} Would you agree that raising the paperweight eight feet off the ground, would take twice as much effort, or work, as raising it four feet off the ground? And raising it sixteen feet, would take four times the effort as raising it four feet?

{Xena:} I’ll accept that.

{Academos:} This is just elementary physics.

{Philosopher:} Then let us return to the paperweight raised four feet off the ground. In this position, the paperweight has zero velocity, relative to the Earth, correct?

{Academos:} Obviously.

{Philosopher:} Now, when I let it drop, it seems to pick up speed as it falls, and hits the ground with a thud. The paperweight seems to have its greatest velocity at the instant it hits the ground, correct?

{Academos:} And you claim that the force of the paperweight at that point is proportional to the square of its velocity. I still don’t see it.

{Philosopher:} Let us continue our analysis. Notice that the paperweight is back to the exact same position from which it started. This means that the work that I transferred to the paperweight in lifting it, was completely expended, so to speak, when it fell. The net result is “zero”– no change.

{Xena:} Wait a minute. You’re saying that the work required to lift the paperweight, somehow equals the force of the paperweight when it hits the ground?

{Philosopher:} Exactly!

{Academos:} But this is nothing new. Every physics textbook explains how potential energy is converted to kinetic energy.

{Philosopher:} However, my dear Academos, all of these concepts originate in Leibniz’s work on Dynamics.

{Academos:} They do?

{Philosopher:} This is how Leibniz puts it: “Thus there appears a new twofold distinction of forces; viz., one– which I call inert or inactive force” (or what you call “potential energy,” Academos) “refers primarily to the element of force while the motion itself does not yet exist in it but only the tendency to motion, as, for example, the stone in a sling which tries to fly off in the direction of the tangent, even if it is pulled back by the chain which holds it securely. On the other hand, the other force, which I call living or active force” (which is your “kinetic energy,” Academos) “is the usual one which appears in actual motion. An example of inert force is centrifugal force, or gravitational or centripetal force, or also the force which tries to restore a stretched elastic body to its original state. However, active or living force appears in impact–e.g., the force or impact of a heavy body that has been falling for a certain time, or that of a stretched bow which gradually resumes its earlier position–and such an active force arises from an infinite number of constantly continued influences of inactive forces.”

{Academos:} All right, I’ve heard enough of Leibniz. How about the issue of measuring the force by the square of the velocity?

{Philosopher:} Think back to the paperweight in free fall. Do you agree that the paperweight gradually picks up speed as it falls?

{Academos:} Everyone knows that there is a constant rate of acceleration of a body falling in a gravitational field.

{Philosopher:} Do you mean that a body weighing one pound dropped from a height of 10 feet, will hit the ground at the same time as a body of 100 TONS dropped from 10 feet?

{Academos:} If they are both dropped at the same instant, yes.

{Xena:} I’d like to see some proof of that!

{Academos:} But everyone has heard of the famous story of Galileo, who simultaneously dropped a ball of lead and a ball of feathers from the top of the Leaning Tower of Pisa. They both hit the ground at the same time. Moreover, now we know that the acceleration of a body in the Earth’s gravitational field is equal to 32 feet per second squared.

{Xena:} What does it mean to square a second?

{Academos:} In other words, Xena, the velocity of a falling body increases by 32 feet/second every second that it falls. After one second, the velocity of the body will be 32 feet/second. After two seconds, the velocity will be 64 feet/second, etc. Naturally, this is approximate, since it doesn’t take the resistance of the air into consideraton.

{Philosopher:} Thank you, Academos, for you have provided us the knowledge we need to reproduce Leibniz’s discovery. First of all, didn’t we agree that there is a certain inertia which is inherent in things?

{Academos:} I think that we all agreed with my definition of inertia.

{Philosopher:} Then how is it that a falling body constantly INCREASES its velocity? Didn’t we agree that a body will tend to preserve its velocity, unless acted upon by some outside force, and that it takes an effort, or work, to CHANGE its velocity?

{Academos:} Obviously, gravitational force acts upon the body and causes the velocity to increase.

{Philosopher:} Aha! Doesn’t this imply that an accelerating body is accumulating force, so to speak, at a non-linear, geometric rate? Consider that, at each instant, the body is moving with a certain velocity, V. Then, an outside force, like a little “shock,” or an “impetus,” which you call “gravity,” is required simply to overcome INERTIA, that is, to overcome the tendency of the body to remain at the original velocity, V. But, once inertia is overcome, doesn’t it require more force to actually INCREASE the body’s velocity? And isn’t this twofold process, of impetus and increasing velocity, occurring at every instant of the body’s motion?

{Xena:} Wait a minute. This reminds me of what actually happens when two freight cars couple, like the example we were talking about before. When the one in motion collides with the one at rest, it seems to stop for a moment with a violent shake, while making all kinds of noise, as if it were working to overcome inertia first, before they both start to move together down the track. I have seen this happen!

{Academos:} Now I suppose that our Philosopher has a quote from Leibniz that purports to explain the implications of all this?

{Philosopher:} In fact, I do. Leibniz writes “that God created matter in such a way that it contains a certain repugnance to motion, and, in a word, a certain resistance, by which a body opposes motion per se. And so, a body at rest resists every motion, and motion, indeed, resists greater motion, even in the same direction, so that it weakens the force of the thing that impels it. Therefore, since matter resists motion per se by means of a general passive force of resistance, but is put into motion through a special force of action, that is, through the special force of an entelechy, it follows that inertia also resists through the enduring motion of the entelechy, that is, through a perpetual motive force. From this I showed that a unified force is stonger, that is, that the force is twice as great if two degrees of speed are united in a one-pound body as it would be if the two degrees of speed were divided between two one-pound bodies, and thus that the force of a one-pound body moving with two degrees of velocity, is twice as great as the force of two one-pound bodies moving with a single degree of velocity, since, although there is the same amount of velocity in both cases, in the one pound body inertia hinders it only half as much.”

You can see for yourselves, my friends, how this analysis implies that force is proportional to the square of the velocity.

Furthermore, I think that this is what Leibniz has in mind when he writes that “the true quantity of motion over a period of time is ascertained as the integral of the individual impetusus,” or that “the calculation of the motion which extends over a definite time-interval is achieved by the summation of infintely many impetuses.”

{Academos:} Now I feel as if you’re trying to brainwash us with convoluted metaphysical babbling! My mind dissolves into confusion just listening to you! These kinds of elaborate abstractions may impress Xena here, but they are not going to convince any educated person.

{Philosopher:} Don’t give up so easily, Academos! Just try to work through the idea. In any case, perhaps we should proceed, as Leibniz did, to calculate forces through a different method, a posteriori, namely, by calculating the force through the effect produced in using itself up, and see if we achieve the same result. Let us suppose, Academos, that a body weighing one pound, falls for one second before it hits the ground. According to your calculation, its velocity when it hits the ground will be 32 feet/second, correct?

{Academos:} That is correct.

{Philosopher:} Now, answer this question for me: How far did that body fall?

{Academos:} At last, we’re discussing practical science! I would calculate it thusly. Since it started with a velocity of zero, and ended with a velocity of 32, its average velocity would be 32 plus zero, divided by 2, which is 16 feet/second. Similarly, if we consider the velocity at one-fourth of a second, which would be 8, and at three-fourths of a second, which would be 24, the average, again, is 16, and we can make the same calculation for every instant of the body’s fall. Therefore, the falling body would cover the same distance in one second, as a body travelling for one second at a constant velocity of 16 feet/second. In other words, the distance travelled would be 16 feet.

{Philosopher:} Very good! Now, Academos, what about a body weighing one pound, which falls for two seconds before it hits the ground? According to your calculation, its velocity when it hits the ground will be 64 feet/second. Now, how far did that body fall?

{Academos:} Well, since it started with a velocity of zero, and ended with a velocity of 64, its average velocity would be 64 plus zero, divided by 2, which is 32 feet/second, and so on for every instant of its fall. Therefore, it would cover the same distance as a body having a constant velocity of 32 feet/second, which travels for two seconds. In this case, the distance travelled would be 64 feet.

{Philosopher:} Excellent, my dear Academos! Now YOU have proven that Leibniz is correct, and Descartes wrong.

{Academos:} What are you talking about?

{Philosopher:} The body dropping 64 feet has twice the velocity as the body dropping 16 feet, correct?

{Academos:} Yes, but what does that prove?

{Xena:} I see it! It takes four times the work to raise a body 64 feet, than to raise it 16 feet, but only twice the velocity!

{Philosopher:} Xena has the idea. We showed that the force of a body in free fall from a certain height, is equal to the work required to raise it to that height. This means that the force of the body dropping 64 feet, is FOUR TIMES the force of that body dropping 16 feet. But the velocity of the body dropping 64 feet, is only TWICE the velocity of the body dropping 16 feet. This means that force is proportional to the SQUARE of the velocity, because twice the velocity leads to four times the force.

{Academos:} You mean if Descartes were right, the velocity of the body dropping 64 feet, would have to be TWICE the velocity of the body dropping 16 feet?

{Philosopher:} Exactly.

{Academos:} I’m stunned! Now there seems to be no way out of this conclusion! Philosopher, I can see why they compare you to a sting ray! And to think that I did the calculation myself!

{Philosopher:} But now, my friends, you can see more clearly how a tiny body weighing 1/100th of a pound and moving at 1000 miles per hour, if harnessed by technology, can accomplish ONE HUNDRED THOUSAND TIMES the useful work for Mankind, than can a huge body weighing 1000 pounds and moving at only 1/100th of a mile per hour– even though the famous Monsieur Descartes says they are equivalent!

Think, now, about all the tiny droplets of water in high-pressure steam, which move at such high velocity. Think about the force of explosions of gunpowder or gasoline, with such small particles moving so swiftly. Then, consider the microcosm, beyond our senses, containing those “worlds within worlds” spoken of by Leibniz, that infinitesimal realm of “non-linearity in the small.” Think about those infinitesimal worlds in motion, at speeds almost beyond our imagination. All of these wonders await our discovery. They are there, waiting to be harnessed by Man.

As Leibniz puts it, so poetically, “there always remain in the depths of things, slumbering parts which must yet be awakened and become greater and better, and, in a word, attain a better culture. And hence, PROGRESS NEVER COMES TO AN END.”

What a joyful thought! What great reason for optimism!

Let us live our lives accordingly.