# Can Newton's laws of motion be proved (mathematically or analytically) or are they just axioms?

Today I was watching Professor Walter Lewin's lecture on Newton's laws of motion. While defining Newton's first, second and third law he asked "Can Newton's laws of motion be proved?" and according to him the answer was NO!

He said that these laws are in agreement with nature and experiments follow these laws whenever done. You will find that these laws are always obeyed (to an extent). You can certainly say that a ball moving with constant velocity on a frictionless surface will never stop unless you apply some force on it, yet you cannot prove it.

My question is that if Newton's laws of motion can't be proved then what about those proofs which we do in high school (see this, this)?

I tried to get the answer from previously asked question on this site but unfortunately none of the answers are what I am hoping to get. Finally, the question I'm asking is: Can Newton's laws of motion be proved?

• What do you want to prove them from? Asking whether one can prove something is a meaningless question unless one specifies the axioms one is allowed to use in the proof. Nov 12, 2016 at 16:02
• These questions always baffle me. Why would you expect mathematics unsupported by physical evidence to prove the underlying relations of physics? Even if you found a mathematical structure that gave you the results you wanted it's hardly be a week before another article was submitted for publication in a mathematics journal detailing the way varying the assumption gives different results, and you'd be thrown back onto comparison with the real world to decide which to use. And that is the point: physics is descriptive above all else or it is useless. Nov 12, 2016 at 18:38
• @IshanSingh That's clearly impossible, since that system contains only propositions about sets. It does not even mention the real world. Nov 12, 2016 at 19:44
• The links you provide contain no actual proof. In this it is only showed how Newton's second law and law of gravitation agree with observations, this is a PSE question with no proof as an answer, and this is honestly just rubbish. Nov 12, 2016 at 20:46
• I think you are confusing the concepts of mathematical proof and scientific proof. In mathematics, you prove theorems from some set of assumptions (axioms), using the rules of logic. In science, you "prove" a hypothesis by showing it is consistent with observed data and predictive of new data. However, the "scientific proof" concept is misleading, because you cannot really prove scientific theories in the mathematical sense - you can only test how well they explain the observed data. Newton's laws are consistent with many observations, so in this sense they are scientifically "proven". Nov 14, 2016 at 9:17

If you want to prove something, you have to start with axioms that are presumed to be true. What would you choose to be the axioms in this case?

Newton's Laws are in effect the axioms, chosen (as others have pointed out) because their predictions agree with experience. It's undoubtedly possible to prove Newton's Laws starting from a different set of axioms, but that just kicks the can down the road.

• If you take $F=ma$ as your axiom, then $F=\frac{dp}{dt}$ can be proven from that (you need some additional definitions). If you take $F=\frac{dp}{dt}$ as your axiom, then $F=ma$ can be proven. Either statement works as an axiom. Nov 12, 2016 at 17:06
• +1 but I would elaborate on the "kicks the can down the road" part; it is true, but assuming something like "laws of physics don't depend on location, orientation, or time" or "$F = ma$" are two different things. Technically both are axioms but one is a lot more reasonable than the other (Note: I'm not saying that you can derive the latter from the former, just an example :) )
– Ant
Nov 13, 2016 at 13:41
• @Ant The location and time independence of the laws of physics is not sufficient - you also need Nature to obey a relatively constrained Lagrangian formalism, which is just as obtuse as Newton's equation of motion. Nov 13, 2016 at 21:06
• @EmilioPisanty Yes, that's why I said "I'm not saying you can derive the latter from the former"; but my point is that axioms are not created equal, so choosing more general, more abstract and less specific axioms can be great, it's not just "kicking the can down the road", which would suggest that it's more or less the same thing. Don't you agree?
– Ant
Nov 13, 2016 at 23:49
• @Ant Nope, I don't agree. Your axioms might be more general, more abstract, or less specific, but I really don't see how "classical mechanics obeys the principle of least action for a local lagrangian that's a function of coordinates and velocities only, but no higher derivatives" (plus a slew of symmetry constraints) is any more "reasonable" a starting point than Newton's laws - rather the contrary, in fact. You're kicking the can up the road. Nov 14, 2016 at 10:16

In some sense, Newton's Second Law can be "derived" from the assumption that a system's evolution is determined only by its initial position and velocity. This is the argument put forward at the beginning of V.I. Arnold's Mathematical Methods of Classical Mechanics. He starts Chapter 1 with the following "experimental facts":

1. Our space is three-dimensional and euclidean. Time is one-dimensional.
2. There exist a set of coordinate systems (called "inertial") possessing the following two properties: (a) All the laws of nature at all moments of time are the same in all inertial coordinate systems. (b) All coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial.
3. The initial state of a mechanical system (the totality of positions and velocities of its points at some moment of time) uniquely determines all of its motion.

Suppose that our system is determined by $N$ real numbers, which we can assemble into a vector $\mathbf{x}$. Since "experimental fact" #3 says that all properties of the motion are determined by positions and velocities, the acceleration of $\mathbf{x}$ (in particular) is determined by these quantities. We can then conclude that there exists a function $\mathbf{f}: \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R} \to \mathbb{R}^N$ such that $$\ddot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \dot{\mathbf{x}},t).$$ This can be viewed as defining $\mathbf{F}$ for a given system; if we multiply each component of $\mathbf{f}$ by the "mass of each point" (in some appropriate sense), we would get "the force on each point." So the fact that initial positions and velocities determine the motion implies the existence of Newton's equation for some function $\mathbf{F}$.

Note that this implication goes the other way as well. If we assume that this function $\mathbf{f}$ exists, there are theorems from the field of ordinary differential equations that guarantee the existence and uniqueness of the solutions $\mathbf{x}(t)$ to this equation. (In other words, we don't need to define an independent function for $\dddot{\mathbf{x}}$ or some higher derivative to determine the motion; one function that determines the second derivative is sufficient.) Thus, the "experimental fact" that initial positions and velocities completely determine the motion is entirely equivalent to the statement of Newton's Second Law.

• I prefer this answer because it actually attempts to answer the question. Nov 15, 2016 at 5:55
• I would say that this may be one of my favorite answers on the Physics Stack Exchange. While I would also have been quick to parrot the answers discussing the non-uniqueness of Newton's Laws as axioms for Classical Mechanics, when we start with Newton's Laws, the concept of a "force" becomes nebulously relied upon, yet is quite difficult to define both clearly and rigorously for the purpose of analysis. This approach instead depends upon the arbitrariness of frame choice and dimensionality of space and time, which (in my opinion) are more intuitive, especially to the punctilious reader. Bravo. Nov 29, 2016 at 6:27
• @pixatlazaki: Those are some very kind words! But all credit for this line of thinking has to go to that magnificent bastard, V. I. Arnold himself. If you're the kind of physicist who appreciates this type of thinking, the rest of his book would probably also be right up your alley. Nov 29, 2016 at 13:43

To ask for a proof of a law is silly. A law is something which is given to explain a phenomenon. It is valid as long as something does not contradict it and it is able explain things correctly. As far as Newton's laws are concerned, they are already contradicted by Einstein. So it is not valid as the basic axioms used by it like the consistency of time intervals and length in different frames of reference are disproven by the theory of relativity. The fact that it uses Euclidean geometry which is already disproven along with its axioms clearly disproves Newton's laws themselves. Even then it is well and easy to apply in speeds negligible with respect to the speed of light and hence we use it. At last I would say that any law in science does not need a proof. Also if it will be proved, will not it become a theorem?

• The basic axioms are not disproven by relativity, because as others have said, they are simply observations of the way the universe works when you don't go too fast. Einstein just pointed out that the universe works differently when you do go fast. Nov 13, 2016 at 4:25
• It's not silly at all. The entire purpose of physics is to search for ever-deeper relationships of nature that can form the axioms of richer theories from which older laws become emergent phenomena (and which can be proved to be such). We may yet find a new and more descriptive physics that Newton's laws just happen to fall out of.
– J...
Nov 13, 2016 at 12:43
• We just neglect the effect of motion at lower speeds. Relativity applies to all objects. But since the effect is negligible at lower speeds and thus relativistic mechanics and classical mechanics gives approximately same results at this condition. Also axioms like parallel lines intersect only at infinity, sum of all angles of a triangle is pi radians etc and all euclidean geometry is simply based on the observations which are all wrong in respect to relativity theory. Albert Einstein was truly a genius to formulate a theory which contradicts general experiences so drastically. Nov 14, 2016 at 6:46
• @A---B What we observe is not always true. Nov 15, 2016 at 1:20
• At high speeds space-time is not flat as we observe. Suppose making a triangle on a curved surface or meeting of two parallel lines at equator at poles. Albert Einstein once said this to his son upon asking about his popularity "We all are like insects on a very big balloon and I was lucky to notice that it is curved." Nov 15, 2016 at 1:29

Your bigger question is probably: "What is the relationship between physics and mathematics?". Richard Feynman has an excellent talk on that relationship between physics and math. You can find it here: Richard Feynman - The Character of Physical Law - 2 -The Relation of Mathematics to Physics. He begins the discussion on this point at 22:55.

Paraphrasing Feynman feels a bit like heresy, but his point is that even if you were to approach physics axiomatically, there would be a lot of choices you could make on which idea was the axiom and which was the theorem. Geometry has a similar problem. The most pragmatic thing to do then is to have a set of principles that are both useful for working things out, and ones that you have a lot of confidence in. That confidence comes from being simple enough to have clear implications and having checked as many of those different implications as you can.

They are an approximation to General Relativity, so yes, they can be proven using general relativity.

• But General Relativity was set up using Newtonian mechanics as a limiting case. So this is a circular reasoning! Nov 12, 2016 at 22:43
• @freecharly What? I'd like to see that. All the definitions of GR show NM as the approximation, they certainly don't derive from NM. In fact, it doesn't even build on many observations - just the conservation laws and the assumption of laws being the same in all places and times. Of course, ultimately there is circular reasoning - everything in physics is circular, since physics describes reality. There's hardly anything ground-breaking about that, it's true for every science. The only problem is when you count the same evidence twice, and that certainly isn't the case here. Nov 13, 2016 at 19:04
• @Luaan - Of course, Einstein didn't derive GR from Newtonian mechanics. But Newtons mechanics, in particular the equivalence of inertial and gravitational mass, together with his law of of gravitation was used (to get the right form) in the development of GR. There shouldn't be any circular reasoning neither in science nor in any other human thinking. Your statement "everything in physics is circular" is an obvious fallacy. Nov 13, 2016 at 19:32
• @ian - You are, of course, correct that Newtons laws (including gravity's) can be deduced from GR as an approximation under special conditions. But GR was set up so that it would do so. Thus GR is a more general theory with new axioms building upon and replacing Newtons laws including his universal law of gravity. Nov 13, 2016 at 20:35
• @freecharly, Newton's laws can be derived by mathematical proof from GR. The converse is not true. Thus in this context, GR is the axiom; Newton's laws are the derived results. Nov 14, 2016 at 4:48

One can derive the laws of classical mechanics from quantum mechanics. Classical mechanics can be reformulated in terms of the principle of least action. The time evolution of a system is such that a quantity called the action is minimized. According to quantum mechanics, the system will evolve in a probabilistic way, the probability of finding a certain outcome is given by a certain integral involving the action that is over all possible paths. For a system in the regime where classical mechanics should be a good approximation, what happens is that the contribution to the path integral path will be dominated by those paths that are at or near the minimum of the action.

Logic is essentially a tautology -- saying exact same thing twice in different ways. To prove something is the process that ensuring a statement (what you trying to prove) is equivalent to the previous statements (the premise or the axioms).

In this light, asking about a mathematical proof of a fundamental physics law is a bit pointless. We can NEVER be sure a physics law is absolutely right by sole mathematics - all we are doing is just rephrasing something else (sometimes adding a bit extra assumptions) - we can only increase our confidence in it by repeating experiments on it.

Moreover, there are a few tricky points,

Newton's I law: It defines what inertial frames are -- You can't prove a definition, you can only state a definition.

Newton's II law: It is partly a definition, partly an empirical law.

Newton's III law: It is an empirical law.

Empirical laws can be seen as a general declare about our nature - something that can only be disproved by experiments. And by mathematics, we calculate their consequence, or derive them (rephrase them) from other given laws (e.g. F=ma from Lagrange).

ZFC is one of the axiomatic systems we require for mathematics.

In physics however, we need to assume some external conditions which are set in the universe (which may change in other universe). I would not call them axioms, but rather conditions/constraints which can only be found experimentally.

It's like asking what is the color of a ball kept kept in a box, you can only find out by observing it, since it is just an information.

The interesting part is what is the fundamental theorem/law which governs what will be the connection between various physical quantities? Where is all this coded? In other words, why should $V = I R$ and not $V = kIR^3$ for some constant $k$. Until we gain a better understanding, these must be found by experiments.

• These conditions can be named assumptions. But in the theory based on them they play exactly the same role as axioms, so it's good enough to say e.g. that Newton's laws (+Galilean relativity) are axioms of Newtonian mechanics. Nov 12, 2016 at 20:04
• Newton's laws while seemingly play a fundamental role, the further advancement is done using mathematics. The laws can be expressed as mathematical equations which are not anything different from some other mathematical equation. For example, whether you take $F = G \dfrac{m_{1} m_{2}}{r^2}$ or $F = G \dfrac{m_{1} m_{2}}{r^3}$ the mathematics behind the whole theory will be the same. So essentially, these laws are mere unknown assumptions. Nov 12, 2016 at 22:54

tl;dr: No, but basically they're equivalent to the conservation laws of total energy and momentum of a system.

The first law (object has constant momentum unless a force acts on it) is, in my eyes, merely a special case of the second law (since acceleration is per definition the change of velocity and the second law gives its connection to force) - so nothing to prove there except for some thinking about the connection between momentum and velocity.

The second law (change of momentum of an object is proportional to the force acting on it) is basically a definition of force assuming you already know what "mass" is. A definition is nothing you can nor need to prove.

The third law (actio = reactio) is the most interesting one. It states that this "force" defined above, not only happens to stem from an interaction between two (or more) objects (i.e. $\vec F_i = \sum_{j\neq i}\vec F_{ij}$) —which is again just a definition— but that this works in both directions with opposing signs ($\vec F_{ji} = -\vec F_{ij}$). Can we "prove" that?

Not exactly. But let's show it is equivalent to something that (to me) seems quite intuitive:

• Assume no "external" forces (they can be explained away by assuming the objects causing them possess infinite mass).
• If the force is free of vortices ($\vec\nabla\times\vec F_i(\vec x_i) = 0$) and only depends on the object's position, it can be expressed as the gradient of a scalar potential $V_i(\vec x_i)$ such that $\vec F_i(\vec x_i) = -\nabla V_i(\vec x_i)$. The third axiom is then equivalent to requiring that the individual potentials only depend on the distances between the objects, i.e. $V_{ij} = V_k(\vec x_i - \vec x_j)$ (where the index $k$ merely serves the purpose of allowing for different kinds of potential between different objects, e.g. electrostatic or gravitational).
• More complicated dependency of the force involves more complicated potentials, but still boils down to the same thing: All interaction depends on the distance (and optionally relative velocity+).

Via Noether's theorem, the independence of those forces from absolute positions in space (and time) implies the conservation of total momentum (and energy) of all objects together. And that seems very intuitive to me.

+ Note that the force acting on an object $i$ can depend on its position $\vec x_i(t)$ and its velocity $\vec v_i(t) = d/dt\ \vec x(t)$, but not on its acceleration, since that can be fixed by redefining force. Arguing why the force also should not depend on higher derivatives of $\vec x_i(t)$ is more complicated* but also irrelevant.

* I'd start with relativity and metrics only depending on $x$ and $dx$, but that goes too far here...

• That's not a proof, though: the third law posits pairwise action-reaction cancellations, and you cannot get it from global momentum conservation. Further, Noether's theorem requires the universe's dynamics to obey a local Lagrangian formalism with up to first order time derivatives of the coordinates (and that's the 2nd-order ODE character of the second law right there). Why people think this is a more reasonable axiom than Newton's laws is beyond me. Nov 15, 2016 at 10:40
• @EmilioPisanty Nor did I claim it were a proof. Global momentum conservation means the forces involved don't depend on the coordinate origin, so they can only depend on relative coordinates. Ok, theoretically you could assume forces that depend on three or more objects' coordinates (but that would go against Occam's Razor), so conservation alone is not equivalent to Newton's Third. Concerning Noether / Lagrangian, note you could get rid of higher derivatives by declaring them new virtual particles ;) Nov 15, 2016 at 12:16
• But yes, I am handwaiving a bit there... Nov 15, 2016 at 12:17
• I don't really see that the Occam's Razor argument works here. You have a big cloud of $N$ particles all interacting in some wonky way and with a very complex motion, and you observe that the global momentum is conserved: how do you conclude that there are pairwise forces that obey the third law? Moreover, you have to go very deep for this to be true - even molecules don't obey strict pairwise interaction, since A can polarize B and thereby affect its attraction to C. The third law is strictly stronger than the global conservation of momentum. Nov 15, 2016 at 14:05
• On the Lagrangian, declaring new particles with crazy coordinates goes in completely the wrong direction. Say I give you the dynamics governed by the lagrangian $L= a \ddot x^2 + b\dot x^2 +c x^2$, where $x$ and its derivatives are observable quantities - you're claiming that inventing imaginary degrees of freedom just for the sake of keeping a 2nd order ODE is the simpler model? What about lagrangians of the form $L=x(t)x(t-T)$? And what is it exactly about the principle of stationary action that makes it a more "reasonable" starting point than Newton's laws? The razor points to the latter. Nov 15, 2016 at 14:10

I always thought that only the third law is an axiom; and it is fairly intuitive that you can exert a force only "pushing away from something else".

All the rest of classical physics — including the two other laws, and the conservation of energy — follows when one assumes that time and space are the same for all concerned (yes, that may be considered kicking the can down the road ...).

A more abstract wording of this fundamental principle could be: "All interaction is — as the word suggests — mutual, and happens in a common frame of reference."

To be honest, I'm not so sure about the common frame of reference in relativistic physics; although intuitively I would say that this fundamental sentence still holds: the common frame of reference just gets more complicated.

• I agree with your statement and elaborated on that a bit Nov 15, 2016 at 10:19
• Yeah, except that the third law is violated for situations as simple as two electric charges interacting via magnetic forces. Nov 15, 2016 at 10:43
• @EmilioPisanty I beg to disagree. The third law and the conservation laws all hold for the overall system including the fields. Anything else would be surprising because macroscopic forces between colliding bodies -- which, as we know, experimentally observe Newton's Laws and the usual conservations -- are all of electromagnetic nature on the microscopic level. And then there is the question "what is a particle" and "what is a field" -- the distinction in the question seems artificial, if we assume e.g. electrons as charged "particles" ... Nov 15, 2016 at 11:13

I was encouraged to develop my comment into a full answer.

There have been various (very approchable to the novice) papers that develop special relqtivity from first principles, including reports that Galileo could have figured it out without calculus. This is elaborated upon in this answer.

These papers show that given symmetry, further developed from the idea of reciprocity (if A sees B move at velocity X than B sees A move at velocity −X), the general form that velocity addition must have can be determined. This includes Galileo’s fixed time and space as a special case. It’s commonly said to derive special relativity from first principles: so does it also derive Newton’s laws as a special case, or are those assumed as input to the process?

The starting point of assuming symmetries will give you Newton’s First Law: a base case stating that inertial reference frames exist. However, just because you know how velocities add doesn’t mean that you are handed the concept of forces and momentum.

So, you can postulate Newton’s first law, or you can postulate space and time symmetry which is more precise in meaning.

To prove Newton’s other laws you need different axioms to start with; they don’t just appear out of nothing (beyind showing that they are a possible consistent set of rules within the established symmetries).

Other answers here do this. Start with something we now consider to be more fundamental. You need to postulate the idea that objects have different resistance to motion, can exchange momentum etc. and the main law that nature minimizes some quantity, and you can derive the specific formulas for the laws.

Newton's First Law effectively states that momentum is conserved. Newton's Second and Third Laws can be derived from the First Law, for either of two possible definitions of momentum. The derivation uses the Reynolds' Transport Theorem.

The two possible definitions of momentum can be derived analytically, based on one assumption about what relative motion is. The first definition ignores the speed of light. The second definition is the modern relativistic definition. The second definition has been verified experimentally.

• A few years ago, someone wrote a five-page paper that derived the Lorentz contraction relationship using a symmetry argument and the relative motion of two coordinate systems. I know that this paper is available on-line. Does anyone remember where it is? Nov 12, 2016 at 21:06
• @jasper one example free online is nothing but relativity. See my notes in this answer. Nov 13, 2016 at 19:46
• @JDługosz -- Thank you! That is exactly the paper I was remembering. Nov 13, 2016 at 22:59
• Does anyone have a link to a straight-forward demonstration that the Lorentz contraction implies a definition of how momentum depends on speed? Nov 13, 2016 at 23:00
• How "Newton's First Law effectively states that momentum is conserved"? This sounds so wrong to me. Whether momentum is conserved or not is meaningless unless we are talking about $inertial frames$ Nov 15, 2016 at 11:32