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Newton, in his famous book Principia, stated second law of motion as:

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

A more mordern version of second law states, on wikipedia, as

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

$$\mathbf{F}=\frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=\frac{\mathrm{d}(m \mathbf{v})}{\mathrm{d} t}$$

From Newton's law the best I can do to make it mathematical is is to take $F$ as the force applied and $h$ be the 'change in motion'. Then according to the Newton's law: $$ F \propto h $$

If I compare 'my' mathematical expression to that of Wikipedia's I find that $$ h=\frac{d P}{d t} $$

So my question is how do we know that $ h=\frac{d P}{d t} $?
Or more clearly how do we know that what Newton meant by 'change in motion' is 'change of momentum'? And why did Newton not stated the second law as that of Wikipedia's. After all he was 'Newton'. He had invented calculus!

3 Answers3

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Newton defined "quantity of motion" at the beginning of Book I in Definition II.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

Here, "arising from the velocity and quantity of matter conjunctly" means it is the result of multiplying the velocity and mass.

A comment to another question may answer why Newton didn't express his laws more mathematically:

I'm pretty sure that Newton never wrote his law of gravitation in algebraic form, nor thought in terms of a gravitational constant. In fact the Principia looks more like geometry than algebra. Algebra was not the trusted universal tool that it is today. Even as late as the 1790s, Cavendish's lead balls experiment was described as 'weighing [finding the mass of] the Earth', rather than as determining the gravitational constant.

Addendum:

I've been reading some sections of Newton's Principia more closely and I think Newton is using an ambiguous definition for force. Take the following passage from the Scholium following the Three Laws of Motion:

When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time.

The first use of the word "force" in "the uniform force of its gravity" would seem to be the usual definition of force that causes acceleration. The second use, "impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities," seems to refer to a force acting over time that causes a finite change in velocity. Modern physicists would refer to the latter usage as "impulse," which would be expressed mathematically as $J = F\Delta t$ or $J = \int Fdt$ for varying forces.

Given the novelty of calculus at the time, perhaps Newton could not speak confidently of instantaneous actions over infinitesimal quantities of time, so he only spoke of finite intervals, which would also yield to more geometric arguments of motion through space. If Newton had written more mathematically rather than geometrically, he might have expressed the Second Law as either $$F = \frac{\Delta(mv)}{\Delta t}$$ or $$F\Delta t = \Delta(mv)$$ with both of the left-hand sides of the equations being referred to as force.

Mark H
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  • As we know, Newton's second law is an empirical law. Your answer: In its present form it allows the interpretation that quantity of motion is a matter of definition, rather than an emperical law. I think it is worthwhile to state explicitly that it's not a matter of definition. Concepts must be defined in order to be used, but the science must of course be observation based. – Cleonis Mar 21 '20 at 15:55
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    @Cleonis That forces cause acceleration is not an empirical matter. It is how "force" is defined. Quoting Newton's Principia: Definition IV: An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line. All scientific theories are collections of defintions of abstract concepts and their relations. A useful theory must at some point make predictions about the real world that can be put to empirical test. But, the theory itself are definitions and assertions derived from those definitions. – Mark H Mar 21 '20 at 17:07
  • @Cleonis How would you empirically verify Newton's Second Law? How do you measure a force? If you measure the acceleration it causes on an object, then you are assuming the Second Law is true instead of testing it. If you measure something else, like how much a spring compresses, how do you know that what causes a spring to compress is the same thing that causes an object to accelerate? Abstract concepts like force, energy, and momentum can be motivated by empirical observations, but the definitions of these concepts cannot be empirically verified without spiraling into circular logic. – Mark H Mar 21 '20 at 17:12
  • Your position is an interesting one. Here is the philosophy of science that I endorse: experimental corroboration is what allows us to validate a conceptual framework as a whole. With our technology we can steer a space probe to remote celestial objects, perform gravity assists, and so on. If our conceptual framework (second law, gravity) would be circular reasoning then it would only describe itself, it would not stand in any relation to reality. The power of our technology is evidence beyond reasonable doubt that the conceptual framework underlying our technology is grounded in reality – Cleonis Mar 21 '20 at 18:30
  • I do agree of course that the separate components that make up a conceptual framework can, as a rule, not be verified in isolation. It is at the level of the conceptual framework as a whole that you have enough expressive power. – Cleonis Mar 21 '20 at 18:34
  • So that means $h$=$mv$. But $h$ should be equal to $ma$. So that means that by  "arising from the velocity and quantity of matter conjunctly" means product of mass with differentiation of $v$ with respect to $t$. Am i correct? –  Mar 22 '20 at 03:58
  • @Cleonis Einstein in his book The evolution of Physics on page number 33 [ https://b-ok.cc/dl/2626393/b52412 ] said: "Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavour to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. [Cont.] –  Mar 22 '20 at 04:03
  • [Cont.] If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility or the meaning of such a comparison." –  Mar 22 '20 at 04:04
  • @My_name_is_not_available If $h$ is the "quantity of motion," then the rate of change of the quantity of motion is $dh/dt$. This is proportional to the force applied: $F \propto dh/dt$ – Mark H Mar 22 '20 at 04:05
  • @MarkH but Newton's second law speaks of "change of motion" not "rate of change of motion"? –  Mar 22 '20 at 04:06
  • @Cleonis indeed, experiments are the way of proving and disproving concepts in science. But measuring a arbitrary concepts like 'force', 'mass' doest disprove or prove a theory. It is just measurement of a quantity according to a theory. The thing which disprove or prove a theory is the extent to which it could explain natural phenomenas. Newtonian mechanism with its concepts of "mass" and "energy", worked splendidly in explaining nature. On the same page, Einstein said: "It is really our whole system of guesses which is to be either proved or disproved by experiment. [Cont..1] –  Mar 22 '20 at 04:14
  • [cont.... 1] No one of the assumptions can be isolated for separate testing. In the case of the planets moving around the sun it is found that the system of mechanics works splendidly. Nevertheless we can well imagine that another system, based on different assumptions, might work just as well." –  Mar 22 '20 at 04:14
  • Finalizing my exposltion. Quote from section 3.1 of Misner, Thorne, Wheeler: "All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even use concepts." See more quote and more context It is necessary to read the context; comment space is too small to convey this. – Cleonis Mar 22 '20 at 07:11
  • @MarkH upon searching on Google. I found a book by Sommerfeld. In which he address the problem in detail, https://books.google.co.in/books?id=pbs3BQAAQBAJ&pg=PA3&lpg=PA3&dq=The+quantity+of+motion+is+the+measure+of+the+same,+arising+from+the+velocity+and+quantity+of+matter+conjunctly.&source=bl&ots=z8A1Hlys2e&sig=ACfU3U0dViG5vs-UA6Ub9u37w0QODyPnGQ&hl=en&sa=X&ved=2ahUKEwj6i8Tixq3oAhX0Q3wKHYgwARo4ChDoATAFegQICBAB#v=onepage&q=The%20quantity%20of%20motion%20is%20the%20measure%20of%20the%20same%2C%20arising%20from%20the%20velocity%20and%20quantity%20of%20matter%20conjunctly.&f=false –  Mar 22 '20 at 08:32
  • @MarkH I want to say you thank you. 'Cause of you I have arrived at the answer, I was long searching for. Once again thank you. –  Mar 22 '20 at 08:42
  • @My_name_is_not_available I've added a bit to my answer regarding how Newton uses the term "force." Don't take this as definitive. This is just what the writing sounds like to me. – Mark H Mar 25 '20 at 13:13
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Newton stated his law for constant mass, to day we can think of rockets for example with changing mass. For constant mass i.e. dm/dt=0 the two laws are the same.

trula
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In Newton's time, with the means available, it would have been very hard to present specific experimental evidence for the Second Law.


For the moment assume that at ground level gravity is uniform. Then if the second law holds good a projectile will move along a parabola.

To my knowledge it was only many years after Newton that someone actually set up a specific experiment. The setup allowed an object to be released from some height, with a very repeatable horizontal velocity. Therefore as it was accelerated down by gravity the trajectory should be in the form of an inverted parabola. Nails were hammered into a board, along the trajectory. The object would fall right along that curved row of nails each time. To within measuring accuracy, that trace trajectory was an inverted parabola. Of course, this was still a very crude method of corroboration.

In Newton's time the second law was plausible of course.

In the Principia Newton demonstrated that if you use the second law as universal law of acceleration and you use the inverse square law of gravity as universal law of gravity, then you can derive that all celestial bodies will move along Kepler orbits.

That demonstration was tremendously powerful of course. Kepler's laws were known to hold good. The fact that Newton's law of Universal Gravity reproduces Kepler's laws is overwhelming evidence that Newton's law of Universal Gravity holds good.

In a wider sense that applies for everything that went into the calculation of the orbits of the celestial bodies. If a certain set of conceptual building blocks leads to reproducing Kepler's laws, then that corroborates that entire set of conceptual building blocks.

Specifically, if Newton's Second Law would not hold good, then the combination of the Second Law and the inverse square law of gravity would not reproduce the actual celestial motions.


This answer originates from an exchange that I had in the History of Science and Mathematics stackexchange after submitting a question about the origin of f=ma

Cleonis
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  • Sorry. But I don't think this is an answer to the question. You are just saying that Newton's laws works well. Experiments are in favour in Newton within a range of accuracy. But that was not the question, I was asking. Of course Newton's laws work well, that's why he is still in trend. But that's not the point for now. The question was "are Newton's law and wikipedia's statement different? If not, then how are they same?" –  Mar 22 '20 at 04:56
  • @My_name_is_not_available For sure the form in which the second law is formulated by Newton is very different from the modern notation. F=ma expresses a second derivative. Newton opted to present all of the Principia in geometrical form (even though in private he long since had developed his 'fluxion reckoning', which is differential calculus) The geometrical presentation does not lend itself to expressing the concept of a second derivative. What Newton did in the Principia is logically equivalent to F=ma, but for sure the form is different. – Cleonis Mar 22 '20 at 06:37
  • No. I dont agree with you. Newton didn't gave geometric presentation of his laws. They were all in statement form. Yes, I accept that his book is geometric. But I don't accept this that he explained his law geometrically. He formulated laws, in statement form, then using geometric principals he deduced other theorems and all that. And Newton's law and Morden laws are not different. I found a book by Sommerfeld in which he explains it. The online link is: https://tinyurl.com/rm8zowr. If you want a downloadable link, then tell me. –  Mar 22 '20 at 08:47