Newton was describing properties of bodies. Inertia is a property distinct from the property of how force is impressed onto bodies. Still distinct more was how bodies cause a change in the aether (see his book on Opticks) - gravitational mass. The later Equivalence Principle is not really a "Principle" , but a derived characteristic.

There is no redundancy whatsoever.

The first principal or first law postulate the existence of a special class of reference frames called inertial reference frames or inertial frames for short where an isolated body (also pseudo isolated body) is either at rest or moves uniformly in straight line. The second law states that given the existence of such special class F=m.a holds for frames in that class only.

I agree with Dr. George Stoica’s excellent explanation.

interesting question

I learned by reading that before Newton and since Aristotle, there was an old belief which said that to make an object move you must apply a force to it. The fact of saying in the first law that an object can move (at constant speed) without applying force has no other goal than to correct this old conception which moreover "confirmed human common sense". If this explanation turns out to be true, it would mean that Newton himself knew that the first law was a special case of the second.

To Mr Hodge,

Why this difference between the "absence of forces (first law)" and " a force which is equal to zero (particular case of the second law)"?

Why doesn't Newton kill two birds with one stone?

To Mr Hodge and to Mr Barzi

In his first law Newton does not refer explicitly to a referential. This is a deduction that was made subsequently. The word "referential" or the word "inertia" do not appear in the first law. Do not make Newton say what he did not say explicitly.

B. Mohammed-Azizi

Because he is developing a model. First define some terms (body and weight). then form some properties of the bodies. Then do the relationships and predictions.

Dear B. Mohammed-Azizi and Readers,

In order to answer the question posed in this thread, we must recall the original formulation of Newton's laws as presented in his Principia. I will use the English translation of the Latin text (translated by Andrew Motte and edited by Florian Cajori, University of California Press, 1934):

LAW I: Every body continues in its state of rest, or uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

LAW II: 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.

Suppose the motive force acting on a material point (in the arbitrary reference system) converges to zero as suggested by some comments stipulating the equivalence of the second and the first law in the case of zero forces.

Depending on how we converge with the motive force towards zero, we get a large class of different motions in the limit. Which of these motions should be selected as the natural motion alluded in the first law? .

This is a main reason for including the First law (known also as a Galileo Inertia Principle) in the set of axioms of the Newtonian mechanics.

PS.

In order to get some intuition concerning the role of the subsequent derivatives appearing in the Taylor series expansion of a function describing the force during switching it on (or off) please kindly consider:

Article Beyond velocity and acceleration: Jerk, snap and higher derivatives

The examples provided above should clarify further why the first and second laws are not redundant.

Dear Mr. Janusz Pudykiewicz,

If I understand you well, the original formulation of Newton (in Latin of course) is very far from the current formulation (force=mass X acceleration). Newton was talking about change of motion. By change of motion, I think we have to translate by change of speed (in modulus and/or in direction). Moreover, he does not mention the mass?

If the translation you give is accurate, it means that Newton's second law has been put in its present form without Newton's "participation". In this case why associate Newton's first law (which remains authentic) with a second law reformulated by others (who?)?

Once this problem of formulation is solved, we can talk about the problem of limit which is another aspect of the question.

Dear B. Mohammed-Azizi and Readers,

Thank you for your comments. The original version of the second law in Latin is:

"Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur."

The translation published in 1934 is:

LAW II: 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.

(this translation is accurate; the English language translation of the Principia is in general quite good).

The google translator renders:

Law 2: change in motion is proportional to the motive force impressed, and they occur in a straight line where the force is impressed.

(this is less precise)

I agree with you that "the original formulation of Newton (in Latin of course) is very far from the current formulation (force=mass X acceleration)"

I will provide some answers to your historical question tomorrow.

Dear B. Mohammed-Azizi and Readers,

Concerning your conjecture:

“If the translation you give is accurate, it means that Newton's second law has been put in its present form without Newton's "participation"“

I’m certain that some clarification is provided in the following article:

http://www.iosrjournals.org/iosr-jap/papers/Vol13-issue2/Series-1/I13020161138.pdf

It is evident that the credit for casting the second law in mathematical form goes to Hermann and Euler, perhaps even to Euler alone.

Why is Euler's name not associated with the equation expressing the second law of Newtonian mechanics? It is a subject for historians of science. The problem may be similar to the recent debate over the name of a Hubble law, the name has been changed to Hubble-Lemaitre law.

https://www.nature.com/articles/d41586-018-07234-y

https://en.wikipedia.org/wiki/Leonhard_Euler

Dear Mr Janusz Pudykiewicz,

Thank you very much for the clarifications you are giving us.

It is for me extraordinary that the final formulation of Newton's second law is not due to Newton himself. The law written by Newton (in Latin) is expressed ambiguously. He talks about movement, he does not speak clearly, neither of momentum, nor of speed and even less of acceleration. He does not cite mass in this law while he cites it in the law of gravity (I guess the latter has not been reformulated). All this confuses the mind. The formulation "Force = Mass x Acceleration" seems to be due to Euler or Hermann. Newton's first law is a law due to Newton and has not undergone reformulation. It is therefore illogical to compare Newton's first law with the second law reformulated by others (the author or authors, the date of the reformulation are not historically well determined. It also seems that there were three editions of these laws by Newton himself. Finally, Newton used Kepler's empirical laws on the motion of planets to establish the law on gravitation. This law contains the masses of objects which attract and is inversely proportional to the square of the distance between these objects. The question that immediately comes to mind why Newton gives the law of gravitation with masses when he does not mention it in his second law. Unless than the reformulation by others of his second law was in Newton's lifetime and was known to him.All this concerns a historical truth, the details of which have not been reported or simply ignored.

Dear Mr Janusz Pudykiewicz,

Regarding the comparison between the second law (current formulation) and the first, you say that the first law cannot be obtained from the second because it depends on the way in which the force tends towards zero. According to me, if you write F = MxA, if M is different from zero, whatever the way to make the limit evolve towards zero, F and A evolve in the same way and the limit F = 0 implies A = 0 and therefore, the speed is constant when the force is zero. I do not see any problem which can prevent me from asserting in this case that the first law is a special case of the second (version F = MxA).

Apparently, Newton was well acquainted with the notion of mass and defined it as the quantity of matter. In addition, he corrected the old Aristotle belief that he related force directly to speed before Newton corrected this "description" by saying that force is related to change in speed (acceleration). The question is, what prevented Newton from clearly formulating the Second Law in its final form? There are some elements of the historical context of the time and of the various scholars of the time that we do not know.

A formulation of the laws of Newtonian mechanics is unequivocally due to Newton,

The contribution of Euler consisted in recasting the second principle from the language of fluents and flexions to our current notation,

https://www.mathpages.com/home/kmath414/kmath414.htm

The first law can not be derived uniquely from the second law, some arguments are provided in my first answer.

PS.

Reading the Principia is not an easy task if we consider the general methodology adopted by the Author:

“On a very deep level it was natural for Newton to express himself in anagrams, because he seems to have regarded "contrived obscurity" (in Domson's words) as an essential feature of God's design for the world, and Newton adopted this mode of operation in his own work. Recall that he said he had made the Principia "designedly abstruse" (to avoid being baited by "little smatterers" in mathematics).”

(this quote is from the mathpages mentioned above)

The error in the proposition is the the F is multiply defined. This is not good logic. The First F is the inertia force which is NOT a force exerted on a body, but a force of the body to counteract other forces. The Second is an external force on a body. Not the same kind of F as in the first case. Remember the use of forces is to add the different forces to get a net force followed by the motion. Later, electromagnetic forces may be added.

Multiply defining parameter is not good physics or good logic.

- you posed the indeed interesting question, however in the thread the discussion mostly relates to history and ethics.; and in this case, what you, and, say, Janusz Pudykiewicz , write, really has the sense practically mostly in the relation above.

However now – and seems at least in last a couple of hundreds of years – in physics the 3 “Newton laws”– since really Newton indeed had analyzed and systematized the works of Galileo, Kepler, etc., defined the fundamental physical constitutions “Space” and “Time”, [though fundamentally wrongly – see https://www.researchgate.net/publication/342600304_The_informational_physical_model_some_fundamental_problems_in_physicsDOI: 10.13140/RG.2.2.12325.73445/3, however rigorously and effectively – see the link, first passage in sect. “Conclusion”] etc., are titled as are titled.

So, if we return to the physical problem in your question – why the 1-st law exists, whereas that is 2-nd law case when the force is equal to zero, then the answer is as – Newton not only formulated the laws, by these laws he simultaneously defined corresponding physical notions “force”, “mass”, “speed”, “acceleration”, which all people understood and used in everyday practice, however in the laws they became rigorously defined ones, and between these notions the rigorous mathematical relations were established.

At that to define rigorously the variables in the 2-nd law, i.e. “force” and “acceleration”, it is necessary before to define – what do these variables - what change?, i.e. just to define the inertial motion in a singled physical law.

And the inertial motion indeed very essentially differs from any other motions, say, from motions with any accelerations; and in the practice only comparing with inertial motion some analysis of some physical system is possible; and so, say, analyses in physics are made practically only in inertial reference frames, including Galileo and Lorentz transformations are applied only to inertial frames, etc.

Cheers

Dear colleagues! The first, second and third Newton's laws are simply the law of conservation of momentum, which is written in differential form and its extreme cases when the force is zero or the sum of the forces is zero.

With respect!

Dear Mr Janusz Pudykiewicz,

Once again thank you for your response.

Personally, I was surprised to see that F = Ma was not written explicitly originally. Nevertheless, I did not go so far as to doubt that Newton was the founder of classical mechanics. In the last description of Newton, you add psychological traits specific to Newton which allow us to understand the way he has to work and to communicate which makes him a great scientist but a scientist with his own human characteristics all the same. Obviously not all scientists are alike.

Concerning the fact that the first law cannot be deduced from the second, I point out that this kind of deduction is very common (in books and in mechanics courses) and does not concern me particularly. The fact that the first law is a special case of the second seems obvious to me (to me and to many other people). This is so true that Newton was made to say that the first law serves to define Galilean referentials (otherwise it would only be an obvious implication of the second). I am willing to admit that the first law is independent of the second but as I said previously I would like to understand how the case of the absence of force (first law) would be different from a zero force (particular case of the second law )?

Dear Mr Hodge

The first law speaks of force applied to an object.

From:

https://www.gsjournal.net/Science-Journals/Communications-Mechanics%20/%20Electrodynamics/Download/4537

Lex I:

First law of motion in original Latin

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

LAW I.

English translation of First law of Motion

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

As everyone and their cats and dogs know, Newton's first law is due to Gallileo. The inability to define a "straight line" is included in the second law. The third law is a work of genius but it is due to Newton's belief in the supernatural, and continues more logically into stimulated emission by Einstein, much later.

Dear B. Mohammed-Azizi and Readers,

In order to understand a relation between the Newton laws in their original form I would like to suggest consulting pages 64-68 in the "Cambridge Companion" available at:

http://strangebeautiful.com/other-texts/cambridge-companion-newton.pdf

Please consider the following quote:

"The second law states that a “change in motion” is proportional to “the motive force impressed” and adds that this change in motion is directed along “the straight line in which this force is impressed.” Some commentators have added a word or phrase to Newton’s law so as to have it read that the rate of “change in motion” (or the change in motion per unit time) is proportional to the force. This alteration would make Newton’s second law read like the one found in today’s physics textbooks.

Newton, however, did not make an error here. He chose his words very carefully. In his formulation of the second law, Newton was explicitly stating a law for impulsive forces, not for continuous forces."

(page 65 of the Cambridge Companion)

When looking at the first law we can easily see that:

"... a major purpose of the first law is to make explicit the condition under which we can infer the action of a continously acting, centrally directed force."

(page 65 of the Cambridge Companion).

The fact that Newton considered two different types of forces in his formulation is very often overlooked in the simplistic textbook derivations of the first law as a special case of the second law.

Thank you Mr Pudykiewicz for your invaluable help regarding the historical side of Newton's laws without which it is difficult to fully understand what Newton meant.

There are still some amazing things. The fact that Newton distinguished between three types of force. On page 62 of the book you indicate, it is noted:

"It is in the conclusion of Newton's discussion of Definition 4 that he states that there are" various sources of imprinted force, such as percussion, pressure, or centripetal force. " There is a mixture of fundamental force, percussion force (brief shocks) and compressive force (associated today with fluid mechanics).

Another amazing thing is the conception that there are "two forces constantly fighting each other", the forces of inertia and the applied forces.

Page 63 of the book by bernard cohen

Newton then turns to an important example of centripetal force taken from Descartes, a stone swirling in a slingshot. The stone naturally tends to fly off on a tangent, but is held back by the force of the hand, constantly pulling the body. inward towards the center via the cord. Newton calls such a force "centripetal" because "i is directed towards the hand as towards the center of an orbit. And then he boldly asserts that the case is the same for "all bodies which are made to move in orbits". They all tend to fly "in a straight line with uniform motion ”unless there is force. One can note an anticipation of the first law in the statement that if there was no gravity, a projectile or an orbiting body would move in a straight line "with uniform motion". It follows from this discussion that planets moving in orbits must also be subjected to some kind of centrally directed force. "

Personally I tend to believe that the first law concerns the "continuous" force in its modern understanding while the second concerns the impulsive forces (which are today "marginalized" in mechanics).

But today, everything seems clear because: Fdt = dp = mdv and a = dv / dt.

In wikipedia (French version) we can read:

"The original statement of Newton's second law is as follows:

"The changes that happen in motion are proportional to the driving force; and are in the straight line in which that force has been impressed."

In its modern version, it is called the Fundamental Principle of Dynamics (FPD), sometimes called the Fundamental Relation of Dynamics (FRD), and states:

In a Galilean frame of reference, the derivative of momentum is equal to the sum of the external forces acting on the solid"

First remark: Nothing seems obvious in the passage from the old to the modern version of Newton's second law. The concepts of mass and acceleration or momentum are totally absent from the original law.

Second remark: when Newton says that the motion is in the direction of the impressed force, it contradicts the fact that for a circular motion the force (in modern meaning) is perpendicular to the motion (the velocity vector being perpendicular to the centripetal force). Therefore, in the old version of the second law, the meaning of the word "force" differ from our current meaning (this confirms that this meaning is that of impulsive forces). In this case, by trying to make a link between the second and the first law becomes a nonsense.

The modern translation of Newton's first and second laws gives rise to a redundancy that does not exist in the original version

Dear colleagues,

Upon reflection, original Newton's second law which states that:

"The changes that occur in motion are proportional to the driving force and are in the straight line in which this force has been impressed."

can be in the modern formalism in the following way:

When he says in the first law that in the absence of force, the motion remains the same, it can be translated by the fact that the velocity remains the same (we disregard the mass). If we apply an impulsive force; the velocity before the application of this force and the velocity after the application of this force will have changed by a quantity (vector) dv, the second law says that dv is proportional to the force F (vector) if we call the proportionality factor b, we will have quantitatively: dv=bF. Concerning the direction of the force and the changes of speed it means that the change of speed (dv) is colinear to the force F. It is a mistake to translate Newton's sentence that the new speed is in the direction of the force, it is rather the change of speed dv which is in this dv which is in this direction. The formula dv=bF is no other than F=ma.

Although we have more means today (infinitesimal calculations, vector representation, physical models ....) the phraseology used by Newton remains very rigorous. It is the ease with which his laws are presented to us today that makes us not want to make any effort to analyze. Now, everything becomes formulas and letters in mechanics. If great scientists did not contest his discoveries, it is because they understood that they were fundamental.

Mr Alexander Braginsky,

The principle of conservation of momentum states that for an isolated system, (without external forces) the momentum is constant.

The quantity of motion is defined by p=mass x velocity. Newton's second law is written in its modern version F=dp/dt. It means that for a non-isolated system, this momentum varies (and therefore is not conserved) and this variation with respect to time is equal to the external force applied on the object or the material point.

If you have a different point of view, I am ready to discuss it and maybe adopt it.

There is an other article on the concept of force for Newton which seems different from the present concept and on which Newton's laws were built:

https://www.jstor.org/stable/2708161

The author of "Newton's Concept of Motive Force" writes in particular:

"I wish to draw attention to a remarkable and persistent error in the interpretation of Newton's second Law of Motion, to show the bearing of this error upon Newton's concept of motive force, and to suggest, in the light of this, a possible origin of Newton's three fundamental Laws of Motion"

Another mystery at Newton. Why speak and write in English and formulate mechanics in Latin terms? It would have been easy to write the equivalent of these laws in English (even in old English for us). This would have avoided the difficult problems of translation from Latin.

Another historical fact

From:

Article La traduction et les commentaires des Principia de Newton pa...

Translation of Newton laws from Latin to French by Émilie du Châtelet:

"La traduction (en français) et les commentaires des Principia de Newton par Émilie du Châtelet (1706-1749)

L’apport théorique de Newton

Avec une formulation plus moderne

Elle présente dans l’Introduction-XII à XVII le contenu des Principia et y résume l’apport théorique de Newton. Le Premier livre développe la théorie de la gravitation avec ses solutions générales. En Introduction-XIV, Madame du Châtelet énonce les trois lois du mouvement établies par Newton, enseignées aujourd’hui encore au lycée:

1°. Que tout corps persévère de lui-même [en l’absence de force] dans son état de repos ou de mouvement uniforme en ligne droite.

2°. Que le changement qui arrive dans le mouvement est toujours proportionnel à la force motrice, et se fait dans la direction de cette force.

3°. Que l’action et la réaction sont toujours égales et contraires. (p. 8)"

Retranslation of the above text in English:

"The translation (in French) and commentary of Newton's Principia by Émilie du Châtelet (1706-1749)

Newton's theoretical contribution

With a more modern formulation

In the Introduction-XII to XVII, she presents the contents of the Principia and summarizes Newton's theoretical contribution. The First Book develops the theory of gravitation with its general solutions. In Introduction-XIV, Madame du Châtelet states the three laws of motion established by Newton, which are still taught today in high school:

1°. That every body perseveres of itself [in the absence of force] in its state of rest or uniform motion in a straight line.

2°. That the change that occurs in the motion is always proportional to the driving force, and takes place in the direction of this force.

3°. That action and reaction are always equal and opposite. (p. 8)"

Comment:

The first law state that in the absence of forces that motion is at rest or uniform, i.e. dv=0 (v=velocity in vector)

The second law states in presence of force, that change in motion is proportional to the force, i.e. dv=kf, were f is the force (in vector) and k is the a constant of proptionality.

Thus if f=0 (in the second law) , we therfore have dv=0 (first law). We recover the first law as a particular case of the second law whereas newton separates clearly the two cases of absence and presence of forces.

Dear B. Mohammed-Azizi

The question of Newton's laws as it emerges from your excellent comments is purely historical. After consulting my favorite source of information on physics which is the Landau & Lifshitz course, I noticed that there is not even a single mention of Newton's laws in the first volume devoted to mechanics; we only have a reference to Galilean relativity.

Dear B. Mohammed-Azizi

Citing the Newton's first law lead to express the translational fundamental principle of dynamics which is logically explained as the sum of the inertia forces plus the dissipative forces plus the static forces applied on a system is equal to the dynamic external forces.

Yes, the first law is a consequence of the second law.

If no force is applied on this system then only gravitation force and support forces plus eventual additional static forces exist in the equilibrium case where the acceleration and the speed are zero.

Dear Janusz Pudykiewicz,

Landau (and Lifshitz) uses the formalism of analytical mechanics and more particularly the Lagrange function. He establishes the form of the Lagrange equations from the principle of least action of Maupertuis.

He defines the inertial reference frame (galilean) as a reference frame where time is uniform, space is homogeneous and space is isotropic. By means of these definitions, he deduced that for a free particle (not subjected to an external force) that the velocity is constant in an inertial reference frame (first law of Newton). He then constructs the Lagrangian of a system of interacting particles. Newton's laws are thus deduced from the principle of least action, also from the definition of the inertial reference frame and moreover from the form of the Lagrangian function for a system of particles without and with interactions. Landau follows a more theoretical, more philosophical and less pedagogical approach which is not within the reach of high school students.

I also learned analytical mechanics with the help of this book which is very small but extremely enriching. It must be said that Lev Landau is not just anyone.

Dear Janusz Pudykiewicz,

On page 9, in section 5 "Lagrangian of a particle system", he talks about Newton's equations.

Dear B. Mohammed-Azizi

Thank you for the useful observations. I have just one comment about the pedagogical aspects mentioned in your post. If we follow the special Feynman lecture on the principle of least action:

http://liberzon.csl.illinois.edu/teaching/FeynmanLecturesOnPhysicsChapter2-19.pdf

the entire subject of analytical mechanics is becoming suddenly clear and natural to follow at a very elementary level.

PS.

Concerning the statement from Landau & Lifshitz: "On page 9, in section 5 "Lagrangian of a particle system", he talks about Newton's equations"

it is important to remember that equation 5.3 on page 9 was not known to Newton; both laws were absolutely essential in the original formulation. This fact explains why in Landau & Lifshitz we read: ".... are called Newton's equations".

Every rule has a beginning and an end

Dear Colleagues.

Indeed, in Newton's modern interpretation, the second law absorbs the first one. However, the modern course of theoretical mechanics is often based on axioms, in which the first axiom is formulated close to the answer of George Stoica. "An isolated material point moves evenly and rectilinearly with respect to the inertial frame of reference".

I would like to clarify the meaning of the axiom. THERE ARE SUCH REFERENCE FRAMES (INERTIAL) in which the isolated material point moves evenly and rectilinearly.

Hi Dr B. Mohammed-Azizi . Newton's laws of motion relate an object's motion to the forces acting on it. In the first law, an object will not change its motion unless a force acts on it. In the second law, the force on an object is equal to its mass times its acceleration. See the link: https://www.britannica.com/science/Newtons-laws-of-motion

Returning to the original question which is:

“How could Newton have stated a first law which is a special case of its second law. What does this redundancy mean?“

I would like to share an article supporting the original opinion that there is no redundancy between the Newton laws despite common assertions to the contrary:

http://www.quartets.de/acad/firstlaw.html

Dear Colleague,

Even though Newton alludes to inertial frames of reference in his first law, he does not say so explicitly. We must therefore make an effort to understand this. Readers of these laws will inevitably fall into two categories. The first will tell you that it does not explicitly talk about referentials and will conclude that there is redundancy. The second, will tell you that a genius like Newton cannot make redundancy of his laws and will conclude that the first law implicitly defines referencials of inertia where these laws can be applied. In my opinion, these are the only two interpretations of the problem posed.

The best answer to the question about the completeness of the axiomatic system of Newton is provided in

“Euler, Newton, and Foundations for Mechanics“

Marius Stan

The Oxford Handbook of Newton

https://philarchive.org/archive/STAENA

I strongly recommend this text for the nearly perfect and enlightening answer to the original question posed in this thread.

Some key elements that are related to the initial question.

Whether we like it or not, the question raised here is essentially linked to the history of Newton's second law which has not been written as it appears today (F=ma). I summarize what I understood by reading the article given by the following link:

https://perso.imcce.fr/alain-albouy/albouy_Journees2003dec.pdf

Newton is indeed the founder of the theory of classical mechanics despite the fact that some people attribute to him only a synthesis of works and results and concepts found before him, but he is the author without any doubt of the fundamental law of gravitation and of fundamental new concepts in classical mechanics.

Newton stated the famous second law (1687) only in the form of a sentence stated in Latin language with certain difficulties of translation (for us) due to the meaning of words used by Newton;

He is not the direct author of the vectorial formulation F=ma known today. It took half a century (yes, 50 years) after Newton for his second law to be put in the form of F=ma. This last formulation goes back to 1747 and is attributed to Euler, without being very clear

It seems that the "delay" of its formulation is essentially due to the difficulty that the vector formalism had to "impose itself" at that time.

Text translated from :

https://perso.imcce.fr/alain-albouy/albouy_Journees2003dec.pdf

"Half a century of trial and error

In 1740, nobody had yet deduced from the Principia Newton's equation in its current "cartesian" form (F=ma in vectors). This hardly believable observation can be found in Lagrange's "Mecanique Analytique" of Lagrange:

"This is how we found the known formulas of tangential forces and normal forces, which have been used for a long time to solve problems about the motion of bodies animated by given forces. Euler's Mechanics, which appeared in 1736, and which must be considered as the first great work in which analysis has been applied to the science of motion, is still entirely based on these formulas; but they have almost been abandoned since, because a simpler way of expressing the effect of accelerating forces on the motion of bodies.

It consists in relating the motion of the body and the forces which solicit it to fixed directions in space. Then, by using, to determine the location of the body in space, three rectangular coordinates which have these same directions, the variations of these coordinates will obviously represent the spaces covered by the body

by the body along the directions of these coordinates; consequently, their second differentials, divided by the square of the constant differential by the square of the constant differential of time, will express the accelerating forces which must act according to these same coordinates; thus, by ' equating these expressions to those of the forces given by the nature of the problem, we shall have three similar equations which will serve to determine all the circumstances of motion. This way of establishing the equations of motion of a body animated by any forces by reducing it to rectilinear motions is, by its simplicity, preferable to all others; it should have should have been presented first, but it seems that Maclaurin is the first one who used it in his Traite des fluxions, which appeared, in English, in 1742; it is now universally adopted." " (End of citation)

Dear B. Mohammed-Azizi

Thank you for your excellent summary. It is useful to consider the outstanding historical research of Michael Nauenberg (1934–2019)

https://news.ucsc.edu/2019/07/nauenberg-in-memoriam.html

A well known paper: Hooke, orbital motion, and Newton's Principia is particularly relevant to understand a source of the apparent redundancy:

American Journal of Physics, Volume 62, Issue 4, pp. 331-350 (1994).

Pub Date: April 1994 DOI: 10.1119/1.17576 Bibcode: 1994AmJPh..62..331N

PS.

http://physics.ucsc.edu/~michael/hookeorbitalmotion.pdf

https://www.degruyter.com/document/doi/10.1515/opphil-2020-0131/html

Dear colleagues,

I give you below the title and the summary of another article which endorses in the same subject.

==================================================================

Is Newton’s second law really Newton’s?

Bruce Pourciau

Department of Mathematics, Lawrence University, Appleton, Wisconsin 54911

(Received 1 November 2010; accepted 9 June 2011)

[DOI: 10.1119/1.3607433]

When we call the equation f=ma “Newton’s second law,” how much historical truth lies behind us? Many textbooks on introductory physics and classical mechanics claim that the Principia’s second law becomes f=ma, once Newton’s vocabulary has been translated into more familiar terms. But there is nothing in the Principia’s second law about acceleration and nothing about a rate of change. If the Principia’s second law does not assert f=ma, what does it assert, and is there some other axiom or someproposition in the Principia that does assert f=ma? Is there any historical truth behind us when we call f=ma “Newton’s second law”? This article answers these questions.

VC 2011 American Association of Physics Teachers.

=======================================================

Personal comment:

Thus, the question posed in this discussion inevitably leads to the following question:

Is Newton's second law in its present form the work of Newton? Is it a more or less obvious deduction from the principia? Or is it a work foreign to Newton?

According to the article mentioned above, it seems that Newton reformulated his second law several times. This makes us think that Newton had difficulties to translate his thought in a clear way. If it passed a half century between the original formulation of this law and the formula F=ma. It is because things were not obvious.

Finally, there are many articles on the historical aspect of Newton's laws and the relationship between the second law with the fundamental formula F = ma.

All this shows once again that the translation and interpretation of Newton's notions of movement, force, quantity of movement remain until today subject to hot debates.

Here is the summary of another reference:

=============================================================

Acta Mech

https://doi.org/10.1007/s00707-017-2074-2

NOTE

R. Lopes Coelho

On the deduction of Newton’s second law

Received: 21 April 2017 / Revised: 27 September 2017

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Abstract

We learned at high school and university that Newton’s second law is F = ma. However, Newton never wrote this equation. Furthermore, there is no agreement among historians of science as to the equation that expresses Newton’s second law. On the other hand, Euler claimed to have discovered a principle of mechanics, which is F = ma. The respective paper of Euler provides us with the means of clarifying the issue. We can compare Newton’s second law and Euler’s principle with each other and verify whether there are significant differences between both laws. This is the task of the present paper.

========================================================

see also:

Isaac Newton, Leonhard Euler and F = ma

from:

Article Isaac Newton, Leonhard Euler and F = ma

Because of all that has been said about Newton's second law and its exact origin, it might be more accurate to call it Newton-Euler's law than Newton's second law.

The redundancy in question (in the title of this discussion) does not appear in Newton's original laws. This is now clear.

It is therefore a historical quid pro quo which is at the origin of this problem.

The question that naturally arises in this discussion is: Why do we attribute the formula F=ma due to Euler to Newton (who did not state it this way in his second law)?

Don't we say that we have to give back to Cesar what belongs to Cesar?

AH! finally - a bit of light. Newton's breakthrough in Principia and Opticks has been changed in modern perceptions..

Let's say, to be fair, that if we compare Newton to a hunter and the formula F=ma to a prey, Newton would have aimed right but he only hurt the prey.

B. Mohammed-Azizi

Newton First answer how an object moves when there is no force (interaction), then talk about what happens when there is interaction, and then introduce the specific form of interaction to explore how an object moves under a certain type of interaction.

Newton's first law and second law can be combined completely.

In a static state, Newton's universal gravitation is also correct.

Preprint Gravitational Fields and Gravitational Waves

First of all, it is necessary to look at what Newton really said. An early translation of his principia into English from 1846 can be found at https://archive.org/details/IsaacNewtonPrincipiaEnglish1846/page/n9/mode/2up

On page 83 (page 89 of the PDF) one can find the axioms, or laws of motion:

I. "Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."

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

III. ...

First, the 2nd law is not F = m x a; it is sometimes interpreted as F = dp/dt.

Then, since the alteration of motion is proportional to the force impressed, no force means no alteration. This is equivalent to the statement of the first law. However one should not think that then the first law is not needed. Mathematically, looking at the laws written down as formulas, the first law could be skipped. The crucial thing is however that the different laws have different meanings, or focus. The first law makes a statement on the motion if there is no force at all, meaning if a body does not interact with any other body. Basically it says that if there is no interaction then nothing changes. The 2nd law then makes a statement that is quantitative, saying something about the magnitude of change in motion as a result of the magnitude of the force acting.

So, from a mathematical point of view the first law could be skipped, however physically it is really important.

I would like to point out to Mr. Harald Mehling, that the formulation in the mathematical sense of the fundamental law of dynamics F=Ma (second law) is not due to Newton, but was attributed to Euler and that between the formulation of Newton's laws in the form of text and the mathematical formulation F=Ma of Euler, a considerable time (50 years) has passed.

In my opinion, the two formulations should be disconnected, because from the historical point of view, there is very little to compare (translation from Latin, definition of force in Newton's sense, absence of vector calculus in Newton's time, etc.).

Besides, how can us link the sentence formulation of Newton's second law with Euler's vector formulation, since the two formulations are separated by 50 years. So it was not easy to go from Newton's law to Euler's mathematical formulation (which seems obvious to us today).

It is the fact of wanting to mix the two representations that creates the indicated problem.

Indeed in the formulation of Newton, the first law of Newton does not appear as a particular case of the second law whereas in the mathematical Eulerian formulation it is.

(For more details, please refer to my two answers of August 27, 2021).

Check https://www.wikidoc.org/index.php/Newton%27s_laws_of_motion

The second law builds on the first law and that is why the two laws have withstood the test of time.

Happy New Year 2022 to all RG colleagues.