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Can Newton's 1st and 3rd laws be assumed given just $F=ma$. I know that the argument would be, "No, then there would only be 1 law". But I can't think of any situation where 1 and 3 aren't superfluous.

If you just told me $F=ma$: I would assume nothing else causes an acceleration besides a force. So things not experiencing a force don't change velocity, even when velocity is 0. 1 ✔️

And, when two things that exist interact they use only their mass and acceleration to do so so they both must change in opposite ways. 3 ✔️

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The position you are taking seems to depend on hindsight. Put yourself in the position of Newton being the first person to state these laws.

The first law was a flat-out statement that Aristotle was wrong when he stated that "nothing moves at all, unless a force which causes it to move is acting on it." Of course everybody now "knows" that Aristotle was wrong about that, so the "shock and awe factor" of Newton building his entire argument from that starting point no longer exists.

The second law then gives a definition of how to numerically measure the notion called "force." Of course it is consistent with the first law, since common sense would say that "no force" must have the measured value of $0$.

In modern terminology, the third law is a statement of the principle of conservation of momentum. It is independent of the first two laws - and apparently, the many crackpots who are still trying to invent perpetual motion machines and "free energy" devices still don't believe it is true, despite the empirical evidence (not to mention Noether's theorem).

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    $\begingroup$ The first law is a definition of the intertial frames, no claim on change of motion is given - see my answer above. $\endgroup$
    – gented
    Aug 4, 2016 at 9:03
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    $\begingroup$ Poor Aristotle - one of the most brilliant minds to ever walk this planet and simply happened to arrive too early to be correct and thus we remember him as "That guy who tried to do physics and was wrong." $\endgroup$
    – corsiKa
    Aug 4, 2016 at 15:02
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    $\begingroup$ @corsiKa I think it's fair to say that's the case for all physics. Someone tried to do it before, got stuff wrong, and then somebody else saw the problems and figured out a way to work them out. This is science at its best: a continual refinement of our understanding of the normal behavior of our universe. If Aristotle hadn't tried the wrong ways first, someone else would have, only later. $\endgroup$
    – jpmc26
    Aug 4, 2016 at 15:31
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    $\begingroup$ @PyRulez Definitely. Einstein is also wrong, and so are the scientists who developed quantum theory. Each of those theories has problems as well. But this only strengthens my point: we are ever refining our understanding of the universe. It's just a question of, "How wrong?" and, "Under what conditions are these models still useful?" And in hundreds of years, people might think of Newton and Einstein in much the same way we do Aristotle. $\endgroup$
    – jpmc26
    Aug 5, 2016 at 22:20
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    $\begingroup$ The other point to be made here is that while it is true that all physics is "wrong" in some sense, there are degrees of wrong, and in fact Aristotle is not 100% wrong. His physics is still good for basic movements of objects in the everyday environment on planet Earth. Objects sitting still on the Earth do not usually move, with respect to the Earth, unless you kick them. It's a reasonable model for simple everyday phenomena, it's just that it's not useful for much more than that. That is, its range of validity is very, very limited. But it's not zero. $\endgroup$ Aug 6, 2016 at 3:46
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Newton's first law defines the inertial reference frames:

  • There exist in the universe some very particular reference frames such that, in those very particular reference frames (and only in those and no more) a body not subject to external forces or interactions moves with constant velocity.

Newton's second law states the change in motion in the above defined reference frames (and only in the above)

  • In the above defined reference frames (see law 1) a body subject to external forces behaves as $$\textbf{F} = \dot{\textbf{p}}$$

I don't see how law 1 is a particular case of law 2, as law 2 is only valid after law 1 defines the inertial reference frames.

If you just told me F=ma...

the above is only valid in the reference frames defined by the first law.

And, when two things that exist interact they use only their mass and acceleration to do so so they both must change in opposite ways.

One does not know in principle how two things interact with each other. In particular that they must both change in opposite ways is a non-trivial statement. There is no a priori reason why it should be so (it could be anything else).

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    $\begingroup$ I fail to see how the first law defines the inertial reference frames. Perhaps you could quote it and show how it is a definition of such? Thank you. $\endgroup$
    – dotancohen
    Aug 4, 2016 at 10:54
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    $\begingroup$ @GennaroTedesco Do you have a citation for that? The usual statement of the first law is "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force." ref $\endgroup$
    – alex_d
    Aug 4, 2016 at 12:50
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    $\begingroup$ I've never seen the first law stated like that. It is usually some variant on "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force" or "When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force". I'm unable to read the original Latin, though. $\endgroup$
    – dotancohen
    Aug 4, 2016 at 12:51
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    $\begingroup$ Newton himself could not possibly have stated his first law in terms of inertial frames, because the concept of inertial frames came later. You are perhaps correct to say that the first law can be understood to define inertial frames, but insisting that that is the only correct reading is both ahistorical and, IMNSHO, a poor pedagogical approach. $\endgroup$
    – zwol
    Aug 4, 2016 at 15:17
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    $\begingroup$ @zwol Insisting to state Newton's laws as 5 hundreds years ago is in my opinion un-scientific. The content of the first law is to define the framework for the second law, whatever words one wants to use to describe them, that is the understanding of the law of motion. $\endgroup$
    – gented
    Aug 5, 2016 at 7:10
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No, the three laws are independent.

  • The 1st law does not follow from the 2nd

The point here is that you have to understand what Newton meant by "force". For Newton, fictitious forces, i.e. forces which arise in accelerating reference frames, are not forces.

For example, if you are in an accelerating car, you will experience an acceleration in the direction opposite to that of the car's. But this acceleration is not caused by a (Newtonian) force.

So, for Newton, force implies acceleration, but acceleration does not imply force.

In modern terms, the first two laws would be formulated in the following way:

First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Second law: In an inertial reference frame, the vector sum of the forces F on an object is equal to the time derivative of its momentum: $\vec F= \dot{\vec{p}}$.

Notice the first words in the statement of the second law: in an inertial reference frame. But what is an inertial reference frame? It is that which is defined by the first law. So, in modern terms, we would say that the first law defines the inertial reference frames, while the second law tells us how motion (momentum) and force are related in such frames.

Could we include fictitious forces in the second law to get rid of the first? Maybe. But this is not how Newton formulated the laws, and could result in a lot of complications.

For a nice discussion, see this article.

  • The 3rd law does not follow from the second, either

If we consider a system of two point masses on which no external force is acting, we have, from the 2nd law (let's drop the vector notation for simplicity):

$$F=\frac{d}{dt} (p_1+p_2) =0 \to \frac{dp_1}{dt}+\frac{dp_2}{dt} = 0\\ \to F_1 = - F_2$$

Let's indicate with the notation $F_{ij}$ the force caused on particle $i$ by particle $j$. Since there is no external force, the force acting on particle 1 can come only from particle 2: $F_1=F_{12}$. The same is true for particle 2, so that we obtain

$$F_{12}=-F_{21}$$

So we were able to derive the 3rd law from the 2nd!

...Weren't we?

No. Consider now three particles: we would get

$$F_1+F_2+F_3 =0 \to (F_{12}+F_{13})+(F_{21}+F_{23})+(F_{31}+F_{32})=0$$

That is to say

$$\sum_{ij} F_{ij}=0$$

Of course, $F_{ij}=-F_{ji}$ (Newton's third law) is a solution of this equation...but it is not the only one!

So, Newton's third law is not a consequence of the second.

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    $\begingroup$ Yo, that was mad helpful fool. So then did Newton just guess the third law or did he have some reason to believe this was the case? $\endgroup$
    – BoddTaxter
    Aug 4, 2016 at 13:52
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    $\begingroup$ @BoddTaxter I think he just chose the easiest solution (Occam's razor!). Every other guess would be a bit arbitrary to say the least... PS: You can find a nice related discussion here: physics.stackexchange.com/questions/16162/… $\endgroup$
    – valerio
    Aug 4, 2016 at 14:05
  • $\begingroup$ You prove that the second law implies the third by considering pairs and that considering triplets makes the third law consistent with the second law but you conclude that the third does not derive from the second? What kind of logics is that? $\endgroup$
    – user175150
    Sep 8, 2019 at 7:53
  • $\begingroup$ @valerio How is it possible for an axiom to serve as definition? In general axiomatic systems have primitive notions (undefined terms) and axioms. $\endgroup$ Jun 15, 2021 at 19:15
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The first law is a special case of the second law. The third does not follow from the second law. The third law states conservation of momentum. It holds if the system is described by a potential and the potential depends only on the relative positions of the bodies. For example, in one dimension for two bodies, the potential $U$ must be a function only of $x-y$ where $x,y$ are the positions. So for example with a potential $U(x,y) = Cxy$ the forces are not opposite and equal.

But note that no experiments have found violations of conservation of energy or momentum, so if you have such a model, it's a sign that you are throwing away some degree of freedom, some third body (e.g., heat, gravitational pull on the Earth, et.c.).

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    $\begingroup$ "The first law is a special case of the second law" wrong. The first law defines the intertial frames. Once you define the intertial frames then you state the second law only in those frames. I can't believe there's still such confusion. $\endgroup$
    – gented
    Aug 4, 2016 at 8:58
  • $\begingroup$ @GennaroTedesco I don't see why you'd consider it weird - it's just one of those assumptions you get from hindsight. "Of course intertial frames are inertial, duh!" Most people don't realize that before that, inertial frames were not a known and defined concept (e.g. Aristotelian mechanics have no such concept). $\endgroup$
    – Luaan
    Aug 4, 2016 at 13:34
  • $\begingroup$ @Luaan I don't consider it "weird". It is just not what the first law is. $\endgroup$
    – gented
    Aug 4, 2016 at 13:49
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    $\begingroup$ @GennaroTedesco Where does the first law even mention the term "inertial frame"? $\endgroup$
    – user253751
    Aug 4, 2016 at 22:54
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    $\begingroup$ @AntoniosSarikas Yes, of course. The point I want to make with my remarks to this answer is that it is false that first law is a special case of the second law. The second law only holds in inertial systems, which in turn are defined by the first law as those special systems where a body not subject to external forces moves in a straight line. This is how the laws of dynamics are to be intended and I find puzzling that some people still insist on nitpicking on Newton's definitions of 500 years ago. $\endgroup$
    – gented
    Jun 16, 2021 at 7:49
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Newton's first law, a body remains in a constant state of motion or rest unless acted upon by a force, says that the proper reference frame for observing physics is an inertial frame. If you were on an accelerated frame then objects outside that frame would appear to accelerate without any measurable force. It is of course the case that if a force does interact with a body $F~=~ma$ tells you how that happens.

Newton's third law of motion, the change in momentum on one body is equal in magnitude and opposite in direction to that of a second body when they interact. This interaction can be a contact collision, a field or a spring or other mechanism. This tells us that $F~=~ma$ acts in space isotropically. We might say it tells us that space is isotropic. When combined with the first law it also tells us that a body changes its state of motion anywhere in space, so space is homogeneous. The reason is that a body can be in a constant state of motion, translating its position in space, and the second and third laws operate anywhere a force is present on that body. In a Noetherian sense the third law of motion gives us conservation of momentum.

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This might add something to the preceding answers' discussion concerning the redundancy of the first law.

The two first laws, which relate the change of momentum of a body with the force applied to it, can apparently be summarized in a single equation:

$$\textbf{F} = \dot{\textbf{p}}$$

The first law is then just the special case in which $\textbf{F}=\textbf0$, $\textbf{p}=const$. This arguable tautology was pointed out by the physicist and philosopher Ersnt Mach in his book The Science of Mechanics, A Critical and Historical Exposition of its Principles:

Definition IV defines force as the cause of the acceleration, or tendency to acceleration, of a body. [...] We readily perceive that Laws I and II are contained in the definitions of force that precede. According to the latter, without force there is no acceleration, consequently only rest or uniform motion in a straight line. Furthermore, it is wholly unnecessary tautology, after having established acceleration as the measure of force, to say again that change of motion is proportional to the force. It would have been enough to say that the definitions premised were not arbitrary mathematical ones, but correspond to properties of bodies experimentally given.

The two first laws are, expressed succinctly:

  1. A body will remain at rest or move with a constant velocity1 unless acted upon by a force.

  2. Force is equal to mass multiplied by acceleration.

Mach's argument appears to be: one could express both of them in the single statement "If there is no force then there is no change in velocity" (which is just a consequence of the second law and the definition of acceleration). Then, a change in velocity can only occur if a force is present, but this is just what the first law tell us... then, it would seem that the first law is redundant. This view is also apparently shared by Harald Iro in his book A Modern Approach to Classical Mechanics.

However, I think there's a sense in which the first law can be considered an independent rule. I will quote a fragment of the discussion in this page, as I couldn't explain it better myself.

The key to unmasking the deception lies in understanding what Newton meant by "force". [...] For Newton, force is intimately connected with the frame of reference (or co-ordinate system) in which acceleration is measured. This leads to an important asymmetry: a force will cause an acceleration but an acceleration might not necessarily be caused by a force. An object can appear to accelerate when, in reality, it is the reference frame which is accelerating. For example, if I am seated in a train compartment then this is my reference frame. When the train leaves the station, then from my reference system, it is the train station that is accelerating away although, of course, no force is acting on the station.

If the reference frame is accelerating then a body otherwise at rest will appear to be accelerating away. Only in a framework which is stationary or moving with a constant velocity will a body remain at rest or move with a constant velocity unless acted upon by a Newtonian force. In all other frameworks a body will accelerate even when no force is present.

But the above phrase "a body [will] remain at rest or move with a constant velocity unless acted upon by a Newtonian force" is exactly Newton's first law. Therefore the first law defines the frames of reference in which Newton's concept of force is valid. They are frames of reference in which a body remains at rest or moves with a constant velocity unless acted upon by a Newtonian force. Such reference frames are called inertial frames of reference. All of Newton's three laws involve his concept of force so all three laws are only properly defined within inertial frames of reference.

I strongly recommend reading the rest of the article.

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There's a modern re-interpretation of the first law in terms of differential geometry.

Here, velocities live in the tangent space and accelerations in the double tangent space. However, not arbitrary vectors of the double tangent space are valid accelerations - they neeed to be 'second order': If you think of the double tangent space as coordinated by $x, v, \dot x, \dot v$, we need to impose the condition $\dot x = v$. This means the zero vector of that space ($\dot x, \dot v = 0$) is in general not a valid acceleration.

A covariant connection can be used to lift velocities to corresponding 'zero' accelerations, and the first law can be understood as stating its existence. In non-inertial systems, the connection is non-trivial and responsible for the occurrence of 'fictious' forces.

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The laws can be thought of as providing a framework on which dynamics can be hung. The relation that defines velocity in terms of position gives you the "kinematic law" in that framework, while the relation that defines force in terms of momentum gives you the "dynamic law" in the framework.

By themselves, without further details, these carry little empirical weight. The only assertion is that the description is physically relevant: meaning that the system in question actually can be described in terms of these attributes.

The part, that carries the empirical content and distinguishes the system in question from others that may have the same set of descriptive attributes, is the set of relations that link the momentum and forces to the positions and velocities - the "constitutive relations".

In particular, the laws are meant to also go with the assertion that the force is some function of the positions and velocities, not just a definition of "time rate of change of momentum".

So, in that sense, the description is not tautological. The Laws give you a generic constitution-independent framework for defining laws of motion for the attributes in question and to laying out content, the constitutive relations give you the content, itself.

The chief example is that of relating a body's momentum $𝐩$ to its velocity $𝐯$ and the force $𝐟$ to its position $𝐫$m under gravitational motion, by: $$𝐩 = m𝐯, \hspace 1em 𝐟 = -\frac{μ}{|𝐫|^2} \frac{𝐫}{|𝐫|}.$$ The kinematic law, here, is $𝐯 = d𝐫/dt$ and the dynamic law, $𝐟 = d𝐩/dt$. Together, they are the framework. The constitutive laws for $𝐩$ and $𝐟$ provide the content. In this example, it falls in a class of constitutive relations that can be generated by a single function: $$L = \frac{m|𝐯|^2}{2} + \frac{μ}{|𝐫|}, \hspace 1em 𝐩 = \frac{∂L}{∂𝐯}, \hspace 1em 𝐟 = \frac{∂L}{∂𝐫},$$ which I'll say more about, below.

The whole layout of Newton's Laws was really meant as nothing more than a run-up to Newton's law of gravity. The Principia could just as well have been named A Framework For The Law Of Gravity (And For Other Stuff That It Might Happen To Apply Usefully To, As An Afterthought).

Framing In Terms Of Composite Bodies: Additivity
In the run-up to the laws, Newton frames a picture in which the objects of discussion are (generally) composite bodies. A body is made of sub-bodies. The laws apply to the whole, just as much as they apply to each of the parts.

The mass of a body is framed in terms of the density of the material that makes it up. It is assumed that its parts undergo a convective motion of some sort, so that one can define the momentum of the body in terms of the product of the density and velocity.

The mass is tacitly assumed to be positive; which is required for consistency. Otherwise, you could combine a body with negative mass with one of positive mass to get a composite body of zero mass, which would cause a break-down in the laws of motion.

Mass and momentum are additive. The total mass and momentum of a body is the sum of the masses and momentum of its parts, independently of how the body is partitioned into sub-bodies. It is also assumed that the mass provides some kind of measure of a body's total quantity so that if it retains all of its parts, then its mass remains unchanged.

Tacit, but apparently not explicitly mentioned, is that a body at rest or in constant motion has a moment associated with it (the product of its mass and position); and that this, too, be additive from part to whole. This applies directly for bodies at rest, and by projection for bodies in motion.

In the latter case, of the body's mass is $m$, its position is $𝐫$, its momentum $𝐩$, then its projected moment at a given time $t$ could be taken as $𝐊 = m𝐫 - 𝐩t$. Otherwise, if at rest, it would just be $𝐊 = m𝐫$.

In this way, the relation between momentum and velocity may be recovered. Though the different parts of a body may be moving at different velocities, one could define its average velocity as $𝐯 = 𝐩/m$. Under the First Law, below, an undisturbed body has a constant momentum ($d𝐩/dt = 𝟎$).

If the moment is conserved for the body and the body remains intact; i.e., if $d𝐊/dt = 𝟎$ and $dm/dt = 0$, and if it is undisturbed, so that by the First Law, $d𝐩/dt = 𝟎$, then this implies that $m d𝐫/dt = 𝐩 = m 𝐯$, or $𝐯 = d𝐫/dt$ - the Kinematic Law.

Also tacit in all of this is that there should be many components of momentum that describe a body and its parts as there are components that describe the velocity (and therefore, the position) of the body and its parts.

Framing Of The World: Absolute Rest, Constant Motion, Geometry and Genidentity
The picture of the world presented is that there is an absolute state of rest. This is why the first law is stated in two parts: (1) for bodies at rest and (2) for bodies in motion. The "at rest" and "in motion" was not taken to be relative to some observer, but in an absolute sense.

But along with this assertion is that it should be impossible to determine what is actually at absolute rest; particularly, that all frames that are moving at a constant speed in a constant direction relative to absolute rest should be subject to the same laws as the frame actually at rest, so that no means could be found identify which amongst them was the one actually at rest.

This is why it was necessary (3) to have the second part to the first law; that not only undisturbed bodies at rest should remain at rest, but also (4) undisturbed bodies in motion should continue their motion unabated, lest that be a way to tag the moving body as actually being in motion. So, if you traveled along with such a body in uniform motion, you'd be unable to tell whether you were actually at rest or not.

Why is this necessary? Because without it you have no basis for spatial geometry! The very notion of "point", the very primitive of spatial geometry, itself - in order to have physical relevance - must have duration. But, then: what counts as the "same point at a different time"? For instance, is New York in 2021 at the "same place" at New York in 2001? (And is the mayor of that city in 2001 the "same person" as he was in 2021?) Or is the "same place" at some other part of the Earth that rotated underneath it, or at some other part of the Earth's orbit "where it was" at the time in 2001, or at some other part of the galaxy, "where the solar system was" in 2001.

That quality of being "the same place at a different time" is called Genidentity. The very concept of a "point", in spatial geometry, carries the premise of Genidentity with it - otherwise it is physically irrelevant.

Genidentity is just a back-door way of saying "Absolute Rest". Conversely, "Absolute Rest" is a front-door way of saying "Genidentity". The two concepts are logically equivalent.

Without either, you have no spatial geometry; and without spatial geometry, you're forced to delve one layer deeper, replacing "point" as a primitive by "point at a time" as the true primitive, into a chrono-geometry - space-time.

Galileo, and his "relativity of motion", was the true culprit for that, not Poincaré, Lorentz or Einstein in the future on the eve of the 1900's. They just weren't ready for that in the 1600's or 1700's (but their children would be)!

Interactions Between Bodies: Forces, Their Additivity And ... The Laws.
Disturbances on a body were called forces. Tacitly assumed was that forces were additive. All forces are to be regarded this way: as interactions between two bodies.

They are treated as additive in the sense that the total action by any body on another is to be the sum of the actions of that body on each of the parts of the latter body - no matter how the latter is decomposed into parts.

The First Law: No Self Force
The First Law says there is no self force. A body's interaction on itself is zero; so that in the absence of other forces by other bodies, there should be no disturbance on the body.

The Third Law: Equal And Opposite Forces And Part/Whole Consistency.
The Third Law says that the action of any body on another is equal to and opposite the action of the latter on the former.

The Third Law is the key enabler that allows one to scale up the Second Law - about to be mentioned - from the parts of a body to the whole body. The sum of all the forces acting between the different parts of a body, by virtue of the First and Third Laws is zero.

The Second Law: Dynamic Equation And Constitutive Relations
The Second Law states that the total of all the forces acting on a body, by itself from within itself, and by all other bodies, is equal to the rate of change in the body's momentum: $d𝐩/dt = 𝐟$.

As already alluded to above: this is not meant to be tautological. Going with this law is the assertion that the force be describable in terms of the other attributes of this body and other bodies.

A Generic Example: Reciprocal Action
Here's is one pretty generic case showing how constitutive relations may be defined for a large class of constitutive relations.

Suppose we just lump in all the positions/coordinates of all the parts of an entire system together into $q = \left(q^0, q^1, ⋯, q^{N-1}\right)$, and likewise for its velocities $v = \left(v^0, v^1, ⋯, v^{N-1}\right)$, with the Kinematic Law $$\frac{dq^a}{dt} = v^a \hspace 1em (a = 0, 1, ⋯, N - 1).$$

Suppose we do the same with its corresponding components of momentum: $p = \left(p_0, p_1, ⋯, p_{N-1}\right)$, and the corresponding components of force $f = \left(f_0, f_1, ⋯, f_{N-1}\right)$. Suppose that there is a constitutive relation that links the $p$'s and $f$'s to the $q$'s and $v$'s.

A trivial example is the case covering bodies made up of point-like constituents, where each component of momentum is proportional to the corresponding component of velocity. More generally, one might assume a $p$ versus $v$ dependence that is reciprocal: $$\frac{∂p_a}{∂v^b} = m_{ab} = m_{ba} = \frac{∂p_b}{∂v^a}.$$ In such a case, the matrix $\left(m_{ab}: a,b = 0,1,⋯,N-1\right)$ make up what's called the "coefficients of inertia" for the body in question. The general case presented here allows them to be functions of $q$ and $v$ rather than just constant.

Similarly, we could assume a reciprocal action principle for components of force versus components of position: $$\frac{∂f_a}{∂q^b} = \frac{∂f_b}{∂q^a}.$$ The archetypical example includes the gravitational potential $$U\left(𝐫^0,𝐫^1,⋯,𝐫^{N-1}\right) = -\sum_{0≤i,j<N} \frac{Gm^im^j}{\left|𝐫^i-𝐫^j\right|}, \hspace 1em 𝐟_i = -\frac{∂U}{∂𝐫^i} \hspace 1em (i=0,1,⋯,N-1).$$

Finally, we may have a case where the $p$'s have dependency on the $q$'s, while the $f$'s have dependency on the $v$'s, with another reciprocal action of the form: $$\frac{∂p_a}{∂q^b} = \frac{∂f_b}{∂v^a} \hspace 1em (a,b=0,1,⋯,N-1).$$

From the reciprocal action between the $p$'s, we may infer a function $\bar{T}$ that is dependent on the $v$'s and possibly also on the $q$'s such that: $$p_a = \frac{∂\bar{T}}{∂v^a} \hspace 1em (a=0,1,⋯,N-1).$$ Similarly, expanding on the example just presented, we may assume likewise that a function $\bar{U}$ dependent on the $q$'s and possibly also on the $v$'s exist such that: $$f_a = -\frac{∂\bar{U}}{∂q^a} \hspace 1em (a=0,1,⋯,N-1).$$

Then, substituting into the equation governing the third reciprocal action, we find that $$\frac{∂^2\bar{T}}{∂q^b∂v^a} = -\frac{∂^2\bar{U}}{∂v^a∂q^b} \hspace 1em (a=0,1,⋯,N-1),$$ or $$\frac{∂^2(\bar{T}+\bar{U})}{∂q^b∂v^a} = 0 \hspace 1em (a=0,1,⋯,N-1).$$ From this, it follows that the total $\bar{T}+\bar{U}$ can be separated into functions of the $q$'s and $v$'s alone: $$\bar{T}+\bar{U} = T(q) + U(v).$$ Setting $$V = ½((\bar{T} - T) - (\bar{U} - U)),$$ we can write: $$\bar{T} = T(v) + V(q,v), \hspace 1em \bar{U} = U(q) - V(q,v).$$ Thus, setting $$L(q,v) = T(v) - U(q) + V(q,v),$$ we can write: $$ p_a = \frac{∂\bar{T}}{∂v^a} = \frac{∂(T(v) + V(q,v))}{∂v^a} = \frac{∂(T(v) - U(q) + V(q,v))}{∂v^a} = \frac{∂L}{∂v^a}, \\ f_a = -\frac{∂\bar{U}}{∂q^a} = \frac{∂(-U(q) + V(q,v))}{∂q^a} = \frac{∂(T(v) - U(q) + V(q,v))}{∂q^a} = \frac{∂L}{∂q^a}, $$ and, as a result, the Dynamic Law $f_a = dp_a/dt$ can be written, in conjunction with the Kinematic Law $v^a = dq^a/dt$, as: $$\frac{d}{dt}\frac{∂L}{∂v^a} = \frac{∂L}{∂q^a}, \hspace 1em v^a = \frac{dq^a}{dt}.$$

Thus, these are the classes of constitutive relations that are generated by a Lagrangian $L$; and the role played by $L$ is exactly that: to generate the constitutive laws for the $p$'s and $f$'s in terms of the $q$'s and $v$'s.

More generally, Newton's Second Law was meant to also go with the assertion that some kind of substantial relation between the dynamic quantities - the momenta and particularly the forces - should exist with the kinematic quantities - the positions and velocities. It's not meant as a mere tautology of "defining" $f$ as $dp/dt$ (nor even $v$ as $dq/dt$). Those were only the rasters on the framework, which the structure of the system being described, were to be hung on - that structure being given by a set of constitutive relations.

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