Dot Physics

Many textbooks are pumped up about Newton’s 3 laws of motion. For me, not so much. First, (as many other’s point out) these are really Newton’s ideas about force. Second, the first law is pretty much a special case of the second law. Here are the first two laws (in my words):

Newton’s First Law:

The natural state of an object is constant motion.

Yes. I know that is not how it is normally written.

Newton’s Second Law:

The rate of change of an object’s speed is proportional to the amount of net force on the object and inversely proportional to the mass of the object.

This could also be written as the equation:


I wrote the acceleration and the force as vectors because that is what they are. If you don’t know what a vector is, just ignore the arrow symbols.

How are these the same?

Newton’s 2nd law says that the net force is proportional to the acceleration. What if the net force is zero, or if there is no force? Well, then the acceleration is zero. What happens to an object with zero acceleration? Average acceleration (in one dimension ) can be written as:


Or you could say acceleration is how the velocity changes. If the velocity does not change, the object is either moving in a straight line at a constant speed, or at rest.

So, why the 1st law?

I am going to guess (or maybe I read it somewhere, because I am really not a good guesser) that this has to do with Aristotle. Aristotle had his own laws of motion. They go like this:

  • Artistotle’s First Law: If there are no forces on an object it will stay at rest
  • Artistotle’s Second Law: If you push on something with a constant force, it will move at a constant speed
  • Artistotle’s Third Law: If you stop pushing on a moving object, it will stop. Or…the natural state of an object is to be at rest.

Aristotle didn’t state them like this, and you wouldn’t have to write them as three laws. But that sums them up. You have to understand that Aristotle was THE man. What he said was it. His ideas about force were held as the truth for quite some time. So, I suspect that Newton’s first law is really to counteract Aristotle. Aristotle says the natural state of an object is at rest. Newton says the natural state of an object is constant motion.

Note that Newton wasn’t the only one to come up with these ideas about force and motion, but for some reason he gets the credit. Anyway, the point is that Newton’s first law is rather silly.


  1. #1 Blaise Pascal
    October 6, 2009

    I’ve always preferred the “force on a body = the time rate of change of momentum of the body” form of the 2nd law.

  2. #2 Jean-Michel Courty
    October 7, 2009

    This expression of Newton’s first law is commonly found in textbooks, it is however a simplified version of the actual law which is in fact about the existence inertial referentials.

    ” There exist a class of referentiels where the natural state of an isolated object is constant motion. These referentials are called inertial referentials”

    With this formulation , there are no overlap between the laws :

    Law 1 : existence of a prefered class of referentials where the state of motion of an isolated body remains inchanged

    Law 2 : definition of the force as the origin of changes in the state of motion of a point particle

    Law 3 : reciprocity of action when two particles interact

  3. #3 Fahmid
    October 7, 2009

    The way I learned it was a little different.

    1st Law: There exist reference frames, which we call inertial reference frames, such that an object with no net force will move at a constant velocity.
    2nd Law: In an inertial reference frame, F=dp/dt=ma.
    3rd Law; For objects 1 and 2, if 1 exerts a force F_21 on object 2, then object 2 exerts a force F12=-F21 on object 1.

    So this way, the 2nd Law actually depends on the 1st Law to to prove existence of these inertial reference frames.

  4. #4 Rhett
    October 8, 2009


    Good point. I have never thought of it that way.

  5. #5 Cleon Teunissen
    October 8, 2009

    I cannot resist elaborating on the remarks of Jean-Michel Courty and Fahmid.

    In the case of theory of motion Newton opted to present his results in a form that follows the model of Euclid’s elements. (Several of Newton’s contemporaries did the same in their works.)

    The axiomatization makes the presentation really compelling; Newton’s laws of motion live on to this day. Today, the act of citing Newton’s laws borders on ritual.

    If today we would try to axiomatize theory of motion, would we formulate the laws in the same way as Newton did? In my opinion we wouldn’t.

    First some general remarks.
    It’s rare for physics theories to be axiomatized. For instance, there is not a comparable set of ‘Laws of Optics’. The primary function of the laws of motion is to be evocative.

    Special relativity asserts that space and time are actually spacetime, characterized by the Minkowsky metric. That is, according to special relativity the members of the equivalence class of inertial coordinate systems are related by Lorentz transformations.

    So if today we would try to axiomatize classical theory of motion our first law would have to be all about the geometry of space and time.

    Objects that are not subject to any force will move in straight lines, covering equal distances in equal intervals of time. For each object in unforced motion a cartesian coordinate system can be set up, that is co-moving with that object. With multiple objects galilean transformations relate the inertial coordinate systems to each other.

    Comment on first law:
    The first law asserts that the uniformity of inertia is modeled by the geometry of euclidean space. Inertia is very symmetric; vector addition of position vectors, velocity vectors and acceleration vectors goes according to vectors addition of euclidean geometry. The first law asserts what a straight line is: inertial motion is in a straight line. Lastly the first law asserts a relation between space and time: objects in inertial motion are in uniform motion; they cover equal distances in equal intervals of time.

    Of course, in Newton’t time asserting euclidean geometry (and the validity of euclidean vector addition) would have appeared completely superfluous. But I feel compelled to think that Newton did appreciate that major emphasis must be laid on identifying inertial motion as uniform motion.

    Newton’s second law.

    Comment on second law:
    Newton’s second law has not aged. In relativistic physics, when motion is mapped in a co-moving coordinate system, the second law holds good.

    Third law?
    From a modern point of view Newton’s third law is superfluous; it’s a logical implication of the first and second law.

    Interestingly, many textbook authors choose to emphasize the following: when two objects interact (attracting or repelling each other) then the common center of mass remains in inertial motion. That is, they emphasize that in the course of interactions between objects momentum is conserved.

    Now, momentum conservation is a symmetry principle. Momentum is conserved if and only if inertial space is uniform. Emphasizing the importance of momentum conservation is actually emphasizing the underlying symmetry principle. That is why it is so important that the first law states the symmetries of inertial space explicitly.

    I feel Newton’s formulation of the laws of motion is in serious need of updating to modern knowledge. But there seems to be an awesome reverence for the original.

    Cleon Teunissen

  6. #6 meichenl
    October 8, 2009


    You seem to be jumping around a bit. Conservation of momentum is not an iff relationship to symmetry under translations. That is true only when you also assume a stationary action principle.

    As to the axiomatics, I’m sure you know more about this than I do, but it seems to have serious problems. A theory of physics needs to talk about real things, but in an axiomatic system the fundamental terms are necessarily undefined and meaningless.

    Finally, I don’t see how the third law is a consequence of the first and second. I mean, the third law isn’t even true! If you aren’t careful about accounting for momentum in E&M fields, then charged particles don’t conserve momentum.

  7. #7 Cleon Teunissen
    October 8, 2009


    I was trying to cram a lot of ideas into the text (and even so the posting was longer than is comfortable for a blog comment.) Unfortunately, the squeezing led to an appearance of jumping around. Had I allowed myself more length the narrative would have been smoother.

    In theories of physics any axiomatization cannot be carried through with the kind of rigor that is mandatory in mathematics. Hence my statement that in physics the primary purpose of axioms, laws, principles, is to be evocative.

    Your statement about momentum seems to boil down to: “If I disregard the fact that I have to attribute momentum to electromagnetic fields then for charged particles conservation of momentum will be violated.” Err…yeah…if you disregard known physics then the accounting will run afoul. So don’t disregard known physics.

    Cleon Teunissen

  8. #8 meichenl
    October 8, 2009

    thanks for the clarification about axiomatics

    Re: third law

    It’s just not obvious to me that there are equal and opposite forces, although yes, clearly momentum is conserved if you do the physics right

  9. #9 Cleon Teunissen
    October 8, 2009


    Tempting as it is to discuss these things here, I feel on second thoughts that entries in the comments section of a blog are best confined to responses to the original blog posting. I don’t think it’s a good idea for blog commentators to engage in directly commenting on each other.

    I noticed you have a blog of your own: arcsecond. If you start a thread on your own blog I can reply there.

    Cleon Teunissen

  10. #10 Cleon Teunissen
    October 10, 2009

    Tags: theory of motion, inertia, rant.

    With the benefit of modern knowledge Newton’s first law can be given new meaning.

    In relativistic physics there is a concept of curvature of spacetime; GR spacetime is not uniform. Of course Newton could not possibly have anticipated the 20th century developments, but in retrospect we see a necessity for classical theory of motion to assert that space is geometrically flat. And by implication Newton’s first law does just that. The geodesics of space are all straight lines if inertial space is perfectly uniform all throughout the universe.

    By implication the first law is a statement about a global property of inertial space: perfect uniformity throughout all of space.

    Newton’s second law: the other aspect of inertia is the opposition to change of velocity. Change of velocity is opposed, but not prevented. What would happen in a universe where all particles have zero mass? In such a universe all particles would instantly jump to infinite velocity upon the slightest force. In our universe a subtle balance is struck: acceleration does happen, but the effect of force is delicately mitigated; acceleration is proportional to the impressed force.

    I like to think of the first and second law as focusing on distinct aspects of inertial space:
    – The first law asserting the global uniformity of inertial space
    – The second law asserting F=m*a

    In my next comment I will discuss that the third law follows logically from the first and second law. (That is, if you interpret the first law in the sense that I described above.)

  11. #11 Cleon Teunissen
    October 10, 2009

    Tags: theory of motion, inertia, rant.

    Considering Newton’s third law.

    I want to talk about leverage.
    A landrover that is equipped with a winch can try to rescue a bogged down vehicle by anchoring itself to a tree, and use the winch cable to pull the bogged down vehicle free. It’s the grip of the tree roots in the ground that gives the rescue vehicle the leverage to exert the required force.

    If there are no trees, and the rescue vehicle is a hovercraft, can anything be done? That is, can a hovercraft muster up leverage? As we know the answer to that is ‘yes’. While a hovercraft cannot anchor itself to the ground, a hovercraft’s driver can use the hovercraft’s momentum. If the hovercraft has much more mass than the bogged down vehicle then one or two jerks will do the job.

    Whenever momentum is used to generate force the property that is described by F=ma is capitalized upon. Hammering in a nail: as the hammerhead makes contact with the nail the hammerhead decelerates sharply. The nail is the agent that changes the velocity of the hammerhead, and in doing so the nail is subjected to a force from the hammerhead. The amount of force is proportional to the change in velocity: F=ma

    Let two hovercrafts, (I will refer to them as A and B) be connected by a cable, which is reeled in by A. The force that is exerted upon hovercraft B makes it accelerate at 1 meter per second-squared. As we know, A has the leverage to exert that force upon B if the velocity of A itself changes too. The harder A accelerates, the more force A exerts on B. Of course, all of the above is reciprocal; with equal validity A and B can be interexchanged.

    Both the ability of A to cause acceleration of B, and the acceleration of B itself, are described by the expression F=m*a, and vice versa. So it follows from the very laws of motion that the forces that A and B exert upon each other are the same in magnitude.

    Finally, imagine a simplified Solar system, with just the Sun and the planet Jupiter. Kepler’s laws describe a solar system with a stationary Sun, orbited by planets. Newton’s innovation was to assert that in order for the Sun to exert a force upon Jupiter the Sun must itself be in a state of acceleration: the laws of motion imply that the Sun and Jupiter must both be orbiting; orbiting a common center of mass. (In fact the mass of Jupiter is so large that the common center of mass of the Sun-Jupiter system lies just outside the Sun.) To exert the required amount of centripetal force upon Jupiter, how hard must the Sun accelerate towards the common center of mass? That is described by F=m*a.

    Newton’s third law is logically superfluous: if you take the first and second law as axioms then the third law follows as a theorem.

    I am really curious whether I have been able to get my point across. At worst I have elaborated too much, diluting the message. Please inform me whether the message came across.

  12. #12 meichenl
    October 10, 2009

    Hi Cleon,

    Your argument that the third law is superfluous assumes the third law. For example, if there were no reciprocal forces, the hovercraft could pull anything out of the mud, no matter what, and the hovercraft’s inertia would be irrelevant. This is not the actual state of things, but Newton’s second law doesn’t say it couldn’t happen.

    Solar System: Suppose there is “solar matter” and “planet matter” and the law of gravitation says the solar matter exerts force on planet matter, but planet matter does not exert force on solar matter. Why is that logically inconsistent? It isn’t true, but it logically could be, and it agrees with Newton’s first two laws.

    When you state “Newton’s innovation was to assert that in order for the Sun to exert a force upon Jupiter the Sun must itself be in a state of acceleration…”, that is where the third law is coming in. The second law does not imply that statement. There’s no logical reason I can’t have a possible universe in which I exert a force on something without accelerating. If that’s a property of this universe, it’s a new law.

  13. #13 Cleon Teunissen
    October 10, 2009

    Hi Mark,

    from my point of view your criterium of what constitutes a new law is too lenient.

    Let me make an comparison: in geometry, when the realization came that Euclids fifth postulate can be substituted with other postulates, two other rich geometries were developed: hyperbolic geometry and spherical geometry. Those three exhausted the set of rich geometries. Presumably the fifth postulate can be replaced with more than those three, but they aren’t interesting, as the geometries they they give rise to are trivial or self-contradicting.

    I think that once the principle of relativity of inertial motion is granted, and the second law is granted, then the variations are exhausted, any variation in which the third law doesn’t exist will be either trivial or self-contradicting.

    As seen from my point of view your criterium of what constitutes a new law will categorize every theorem as a ‘new law’, since for every theorem it is logically possible to postulate a universe in which that particular theorem doesn’t hold.
    Such an attitude does not in itself violate logic, but if every theorem is categorized as a new law in itself then the concept of ‘axiom of the theory of motion’ is rendered void of meaning. The concepts of ‘axiom’ and ‘theorem’ have meaning only when there is a criterium to distinguish them.

    Of course I’m aware that all these considerations are judgement calls, not decidable with logic steps alone.

  14. #14 Kaleberg
    October 10, 2009

    Newton’s first law was actually rather important. To start with, it was counter-intuitive, and understanding it was critical, just as its restatement in curved space time is critical to understanding general relativity. From a modern point of view, it is a statement about space-time, just as the second law is a statement about mass. Second, Newton was building a big edifice. His Principia covered a broad range of physical phenomena. He wanted to put each of his early bricks in place carefully so that others could follow his overall structure. Third, Newton was proposing a materialistic explanation of physical reality including the politically sensitive area of astronomy. The first law allows for predictable motion without divine intervention, arguing from a simple case to the more complicated case of planetary motion.

    I agree that there is some redundancy in Newton’s three laws, but he wasn’t trying to build a minimal axiomatic system, he was trying to explain a new, relatively complete, and, at the time, exotic world view.

  15. #15 Cleon Teunissen
    October 11, 2009

    As pointed out by Kaleberg, the primary purpose of the Principia was to be persuasive. I have no doubt that Newton was acutely aware that the Principia would have to be very persuasive.

    As meichenl pointed out, in itself an axiomatic system is purely abstract.

    To define a pure axiomatic system one submits axioms, and the operations with which theorems are derived from the axioms. Like all mathematics, an axiomatic system does not need to be a model of something we encounter in reality.

    Still, the effort of thinking through the process of axiomatization can deepen physics insight

  16. #16 Cleon Teunissen
    October 11, 2009

    About the principle of invariance under time reversal (which is implicit in classical dynamics.)

    In any two-particle system all interactions are invariant under reversal of the direction of time. (If you record a video of a perfectly elastic collision between two spheres then on playing the video you will not be able to tell when the video runs forward or backward.)

    Arguably invariance under time reversal is implicit in galilean relativity. Physical interactions are invariant under mirror reversal, including mirror reversal of the direction of time.

    I think that in a universe without the third law the principle of invariance under time reversal will not hold good.
    Therefore I’m inclined to think that the concept of a universe with galilean relativity, but without the third law, is self-contradicting.

    I wonder: suppose invariance under time reversal is not an implicit aspect of galilean relativity. Then maybe it needs to be included as a necessary element to obtain an exhaustive set of axioms. Then this more complete theory of motion has the third law as theorem.

    (As I understand it, the significance of invariance under time reversal was recognized relatively recently.)

  17. #17 Xavier Terri Castañé
    October 27, 2009

    We need a new principle of inertia, but the Lorentz transformations are not valid for it. First we must define tetracoordinates in a relational way as “the new Lorentz’s transformations” that you can find on Relativity and Cosmology viXra e-prints (see 34, 63 and 64)

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