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Newton's first law is often said to define what an inertial frame is - namely, a reference frame in which a body not acted on by a force will move with constant velocity. In other words, a frame where the first law holds.

However, this raises the question of what is meant by "not acted on by a force" and more generally how and what we define to be a force. Suppose you are floating in space with just one other object nearby. The object is at rest relative to you, and so you determine that your frame is inertial. Now suppose the object starts jiggling about in a seemingly arbitrary way. You have at least these options :

  1. You can say that you are no longer in an inertial frame, but instead being accelerated every which way, and that is why the object appears to move. (Fictitious forces).

  2. Or you can say that you are still in fact in an inertial frame, and the object is now moving because of some time dependent forces that started to act on it. Perhaps even after observing the object's motion for a long time, you would be able to describe the motion, and hence these forces, quantitatively.

Of course when there is more than one object moving around, it might be simpler to choose 1 over 2, for calculation purposes. But if so, then the determination of whether you are in an inertial frame or not is not an objective fact, but rather a subjective convention that you choose to employ for simplification of your calculations. What then, is the objective content of the first law?

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  • $\begingroup$ Great questions. Some might say that this is a duplicate question, but I'd like to jump in beforehand and insist that this question deserves fresh attention due to both its importance and the fact that disagreements among authorities regarding the proper interpretations persist to this day. $\endgroup$
    – David H
    Nov 5, 2013 at 5:23
  • $\begingroup$ It might be helpful to view it from a historical perspective, where we assume the existence of an absolute intertial frame - once you spot it (e.g. fixed stars), then you get your objective meaning. As stated in the comment above, people restate the first two laws to suit the modern perspective in various ways. I think the idea of what a physical theory is and what it's axioms are supposed to do has changed too much to hang yourself up on this. Newtons three laws hence seem a little more mystical compared to e.g. the definiton of how Hamiltonian mechanics (or other modern theories) work. $\endgroup$
    – Nikolaj-K
    Nov 5, 2013 at 8:28
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/13557/2451 and links therein. $\endgroup$
    – Qmechanic
    Nov 5, 2013 at 12:39

3 Answers 3

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What then, is the objective content of the first law?

Newton formulated the first law in this way:

"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." (I.)

I do not think there was any mention of "inertial reference frame" in Newton's book (but I do not know for sure). Newton used the notion of absolute space instead. The first law was mentioned to apply with respect to this absolute space, and by necessity also to all reference frames that are in uniform motion with respect to it.

If we do not accept this setting of the first law (as Ernst Mach, who criticized the notion of absolute space), the first law is not always valid. For example, an observer on a roundabout observes that objects on ground move with acceleration without presence of any corresponding impressed forces.

So in order to preserve the coherence of Newtonian mechanics without absolute space, the notion of inertial reference frame was introduced, which is a notion more close to the principle of relativity. "Inertial frame" is a rigid body such that in its coordinate system the first law is valid.

... the determination of whether you are in an inertial frame or not is not an objective fact, but rather a subjective convention that you choose to employ for simplification of your calculations.

One cannot determine with certainty whether he is in an inertial frame or not. By definition, that would require checking that all free bodies move exactly recti-linearly, which is impossible.

What then, is the objective content of the first law?

In the relativistic view, the content is that the inertial frame exists, or at least, from pragmatic point of view, that frames close to inertial can be found and usefully adopted.

So far the experience shows that the assumption of inertial frame can be always made with success. With varying degree of accuracy, it can be the frame of the Earth, or frame with origin in the center of the Earth-Moon system, or the Sun; all with fixed orientation with respect to the stars. We do not know certainly that such frames are exactly inertial, most probably they are not, but assuming that they are, we obtain very reliable results; their deviation from perfect inertialness is negligible.

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  • $\begingroup$ The wording of the first law quoted in this answer is by Andrew Motte, from his English translation of Newton's Principia. Apparently Newton wrote it in Latin as "Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.". Unfortunately I don't understand Latin. $\endgroup$
    – bdsl
    Feb 10, 2016 at 0:10
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Say you wanted to build a physics model in a computer program. JavaScript is an easy example, here is an object with an x coordinate and a velocity along the x dimension and that is dx. Your program is then something like

object1 = {x: 0, dx: 3} ; 
while (true) 
    object1.x = object1.x + object1.dx; 

So that covers the first law of motion.

Without that, the second law wouldn't really do anything, would it? You could apply force (which changes dx) but without inertia, what's the point?

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  • $\begingroup$ I don't think the language of the analogy is generally accessible, but I also don't think it fits (and I don't mean the fact that you compute infinity). Your $\mathrm dx$ is $x'(t)\mathrm dt$ and if you say the second law is how $x'(t)$ then it also implies that with no force, $x'(t)\ne 0$ in some frame. $\endgroup$
    – Nikolaj-K
    Nov 5, 2013 at 8:20
  • $\begingroup$ Huh? The question was if the first law of motion is substantive. The code I provided is the first law, a body at position x moving at dx per iteration of the algorithm stays in motion at dx. If dx is 0, then it stays at rest. The second law uses force to change dx, but without the first law (x = x + dx) it would literally go nowhere. I would say that makes it pretty substantive. $\endgroup$ Nov 5, 2013 at 15:49
  • $\begingroup$ Space and time are already assumed concepts when we state the axioms. If you speak of $\mathrm dx$ and assign a value to it (and you do in stating the second law as a differential equation which needs initial conditions to be solved), then there is motion (as $\mathrm d x$ is $x'(t)$ in the laws) - independently of the value of the force. At least with the modern notion of defining derivatives in math. I.e. the idea that $\mathrm d x$ is the velocity of a trajectory of something moving though space is needed to write down the second law, and so you can do it without writing down the first. $\endgroup$
    – Nikolaj-K
    Nov 5, 2013 at 17:32
  • $\begingroup$ That's fine but if you want to implement the laws of motion as an algorithm you probably want x = x + dx in there. $\endgroup$ Nov 6, 2013 at 0:04
  • $\begingroup$ This or o version of that which returns a value depenend on the number of iterations you've computed. But I'm not sure who wanted to implement an algorithm, and I don't think it answers the question. not that it's not an interesting approach. $\endgroup$
    – Nikolaj-K
    Nov 6, 2013 at 0:06
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I don't know what you mean by "objective content". But I can tell you what it seems Newton thought the content was. At the time Newton wrote, the first law was a repudiation of the Aristotelian view that force was necessary to maintain an object's motion.

From the Wiki article on Newton's laws:

The ancient Greek philosopher Aristotle had the view that all objects have a natural place in the universe: that heavy objects (such as rocks) wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He thought that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external agent was needed to continually propel it, otherwise it would stop moving. Galileo Galilei, however, realized that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. The tendency of objects to resist changes in motion was what Galileo called inertia. This insight was refined by Newton, who made it into his first law, also known as the "law of inertia"—no force means no acceleration, and hence the body will maintain its velocity. As Newton's first law is a restatement of the law of inertia which Galileo had already described, Newton appropriately gave credit to Galileo.

It seems that, in talking about the first law only being correct in inertial reference frames, you have already accepted the revolutionary content of the first law: the very concept of inertia.

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