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If you consider them as laws, then there must be independent definitions of force and mass but I don't think there's such definitions.

If you consider them as definitions, then why are they still called laws?

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there must be independent definitions of force and mass but I don't think there's such definitions. Suppose I define force as what I measure with a spring scale, and I define mass as what I measure with a balance (which has been calibrated in the same location with a standard mass). –  Ben Crowell Jul 6 '13 at 20:47
@BenCrowell I wish life is that simple. –  metacompactness Jul 6 '13 at 20:59
Just think of the laws as saying "one can assign a 'mass' to each object and a 'force' to each interaction so that ...". It is quite possible that with some laws of physics there would be no consistent assignments of 'masses' and 'forces' making Newton's laws hold. This means that Newton's laws are more than just definitions. For example, Newton's laws imply that an astronaut floating in space can't propel himself without throwing something. Can a definition imply anything? –  Peter Shor Jul 6 '13 at 21:23

3 Answers 3

In my view, standard statements of Newton's laws are usually overly concise, and this lack of detail causes confusion about what is a definition, and what is an empirical fact. To avoid this confusion, let's proceed in a systematic way that makes the distinctions between these definitions and empirical statements clear.

What follows certainly is not the original statement of the laws made by Newton himself; it is a modern interpretation meant to clarify the foundations of Newtonian mechanics. As a result, the laws will be presented out of order in the interest of logical clarity.

To start off, we note that the definitions of mass and force given below will require the concept of a local inertial frame. These are frames of reference in which when an object is isolated from all other matter, it's local acceleration is zero. It is an empirical fact that such frames exist, and we'll take this as the first law:

First Law. Local inertial reference frames exist.

How is this in any way related to the first law we know and love? Well, the way it is often stated, it basically says "if an object isn't interacting with anything, then it won't accelerate." Of course, this is not entirely correct since there are reference frames (non-inertial ones) in which this statement breaks down. You could then say, all right, all we need to do then is to qualify this statement of the first law by saying "provided we are making observations in an inertial frame, an object that doesn't interact with anything won't accelerate," but one could then object that this merely follows from the definition of inertial frames, so it has no physical content. However, going one step further, we see that it's not at all clear a priori that inertial frames even exist, so the assertion that they do exist does have (deep) physical content. In fact, it seems to me that this existence statement is kind of the essence of how the first law should be thought because it basically is saying that there are these special frames in the real world, and if your are observing an isolated object in one of these frames, then it won't accelerate just as Newton says. This version of the first law also avoids the usual criticism that the first law trivially follows from the second law.

Equipped with the first law as stated above, we can now define mass. In doing so, we'll find it useful to have another physical fact.

Third Law. If any two objects are being observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant.

How is this related to the usual statement of the third law? Well, thinking a bit "meta" here to use terms that we haven't defined yet, note that the way the third law is usually stated is "when objects interact in an inertial frame, they exert forces on each other that are equal in magnitude, but opposite in direction." If you couple this with the second law, then you obtain that the product of their respective masses and accelerations are equal up to sign; $m_1\mathbf a_1 = -m_2\mathbf a_2$. The statement of the third law given in this treatment is equivalent to this, but it's just a way of saying it that avoids referring to the concepts of force and mass which we have not yet defined.

Now, we use the third law to define mass. Let two objects $O_0$ and $O_1$ be given, and suppose that they are being observed from a local inertial frame. By the third law above, the ratio of their accelerations is some constant $c_{01}$; \begin{align} \frac{a_0}{a_1} = c_{01} \end{align} We define object $O_0$ to have mass $m_0$ (whatever value we wish, like 1 for example if we want the reference object to be our unit mass), and we define the mass of $O_1$ to be \begin{align} m_1=-c_{01}m_0 \end{align} In this way, every object's mass is defined in terms of the reference mass.

We are now ready to define force. Suppose that we observe an object $O$ of mass $m$ from a local inertial frame, and suppose that it is not isolated; it is exposed to some interaction $I$ to which we would like to associate a "force." We observe that in the presence of this interaction, the mass $m$ accelerates, and we define the force $\mathbf F_{I, O}$ exerted by $I$ on $O$ to be the product of the object's mass and its observed acceleration $\mathbf a$; \begin{align} \mathbf F_{I,O} \equiv m\mathbf a \end{align} In other words, we are defining the force exerted by some interaction $I$ on some object of mass $m$ as the mass times acceleration that a given object would have if it were exposed only to that interaction in a local inertial frame. The problem is that this definition is rather unsatisfying for the following reason. We would like the force to be a quantity assigned to a given interaction that is the same when this interaction acts on another object. Thus far, we have defined force in a way that might, in principle, depend on both the interaction, and the object it's acting on.

To remedy this, we note the following:

Second Law. There exists a class $\mathcal F$ of interactions $I$ such that for any massive objects $O_1$ and $O_2$, the force that $I$ exerts on $O_1$ equals the force that $I$ exerts on $O_2$; $$ \mathbf F_{I,O_1} = \mathbf F_{I, O_2} $$

The second law allows us to unambiguously define the concept of force in the sense that it allows us to assign a certain vector that we call "force" to any interaction. Given an interaction $I$ in the class $\mathcal F$, the force $\mathbf F_I$ associated with $I$ is precisely the force $\mathbf F_{I,O}$ that it exerts on any massive object $O$.

So the second law basically says that there exist certain interactions to which we can associate a vector called "force," and this vector tells us precisely how any object acted on by this interaction will accelerate. The fact that makes this law useful is that the class $\mathcal F$ of such interactions is non-empty. In fact, all interactions we commonly encounter in Newtonian mechanics are in this class, so they can all be assigned a force vector.

As somewhat of a tangent, the following is often also essentially treated as implicit in the statement of Newton's laws.

Let interactions $I_1,I_2, \dots I_N$ in the class $\mathcal F$ (defined above essentially as those to which we can assign forces) be given. These interactions are called superposable provided whenever an object $O$ is simultaneously acted on by all of them as viewed from a local inertial frame, the acceleration $\mathbf a$ of $O$ satisfies \begin{align} \mathbf F_{I_1} + \mathbf F_{I_2} + \cdots + \mathbf F_{I_N} = m\mathbf a \end{align}

Law of Superposition. All interactions in the class $\mathcal F$ are superposable.

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This is my point of view: Newton's propositions are definitions (of course, in your answer mass means the inertia of a point particle). About your last statements:*The existence of local inertial frames*: I think the first proposition (law) defines the inertial frames. Accelerations of isolated masses in local inertial frames are opposite in direction with constant ratio: This is not necessary as proved in schaum's outline of theoretical mechanics. –  metacompactness Jul 6 '13 at 17:41
@metacompactness In step 1, I define a local inertial frame, true, but I also think it's important for the purposes of physics in the real world to make the existence statement. A definition of something without knowing its existence isn't particularly useful. As for the assertion about accelerations of isolated masses; this is precisely what is often called Newton's third law which has been experimentally tested for quite a long time; what exactly do you mean when you say that it's not necessary? –  joshphysics Jul 6 '13 at 17:47
This is how the mass and force are defined in schaum. –  metacompactness Jul 6 '13 at 18:07
+1 for remembering the existence statement. It's a pretty deep thing, really. Plus it's reminiscent of my favorite geometric axioms - for instance, the first axiom of Desargues' geometry is, "There exists a point." –  Chris White Jul 6 '13 at 18:59
@ChrisWhite Agree wholeheartedly. Ironically I didn't appreciate the significance of the existence of local inertial frames in the context of classical mechanics until having learned GR. –  joshphysics Jul 6 '13 at 19:13

I think the answer by Joshphysics is very good. In particular the statement that asserting existence is a key element.

The idea is to restate the laws of motion in such a way that the question law versus definition issue becomes clearer.
In analogy with thermodynamics I will state a 'law zero'; a law that comes before the historical 'First law'.
As with Joshphysics's answer the following treatment is for the Newtonian domain.

Law zero:
(Assertion of existence)
There exists opposition to change of an object's velocity. This opposition to change of velocity is called 'inertia'.

First law:
(The uniformity law)
The opposition to change of velocity is uniform in all positions in space and in all spatial directions.

Second law:
(The acceleration law)
The change of velocity is proportional to the exerted force, and inversely proportional to the mass.

The above statements are not definitions.
For comparison, the zero point of the Celcius scale is a definition; it is interexchangeable with another definition of zero point of temperature scale. The laws of motion are not exchangeable for other statements.

The concept of force is also applicable in statics, hence Force can also be defined in the context of a static case (compression), and then we check for consistency with Force defined in terms of dynamics. As we know: we find consistency.

For mass things are more interesting. Mass is in fact defined by the laws of motion. Trivial example: if you would use the volume of an object as a measure of its mass the second law would not apply universally. It's the law of motion that singles out what an object's mass is: precisely that property for which the second law holds good.

The lesson is that if you would insist that any statement is either a physics law, or a definition, you would totally bog yourself down.

Our physics laws are both: they are statements about inherent properties of Nature, and they define the concepts that the laws are valid for.

Additional remarks:

The first and second law together are sufficient to imply the historical third law. This can be recognized in the following way:

Let object A and object B both be floating in space, not attached to any larger mass.
From an abstract point of view it might be argued: there is a difference between:
Case 1: object A exerting a force upon object B, but B not on A
Case 2: object A and object B exerting a force upon each other.
According to the laws of motion the above distinction is moot. Observationally the two cases are identical, making it meaningless to distinguish between them on an abstract level.

Assume for argument sake that object A exerts a attracting force upon object B, but B not upon A. Both A and B are floating in space. The leverage that object A has to pull object B towards itself is A's own inertia. A has no other leverage, A is not attached to any larger mass. A can pull B closer to itself if and only if A is itself in acceleration towards B. There is no scenario, no observation, where Case 1 and Case 2 are distinguishable, hence Case 1 and Case 2 must be regarded as one and the same case.

The first law and second law together are sufficient to imply the superposition of forces.

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To understand what Newton's three Laws really are, one needs to consider the notion of momentum. Momentum $\vec{p}$ of a point particle is the product of its mass $m$ (which will be defined implicitly later) and its instantaneous velocity $\vec{V}$, so $\vec{p}:=m\vec{V}$. Also, $m \in \mathbb{R}_+$ mass units and $ m:=const $ (reasons are so that $ m$ characterises a particle and does not make vectors $\vec{V}$ and $\vec{p}$ point in a different directions). One also needs to consider the Law of Conservation of a Linear Momentum, which is the consequence of space translation symmetry (contrary to a pupular belief that it is the consequence of Newton's Laws).

Now, let's talk about the Newton's Laws:

Newton's first and third laws: consequence of the Law of Conservation of a Linear Momentum, nothing more.

Newton's second law: a definition of a force, $\sum \vec{F}:=\dot{\vec{p}}$ (which also yields the familiar $\sum \vec{F}=m\vec{a}$)

Remark: a question about measuring masses of point particles may arise, so here is the answer. Consider a system of two point particles moving along the $ x $-axis towards each other. Law of Conservation of Linear Momentum states:

\begin{align}m_1 \left |\vec{V}_{11} \right | - m_2 \left |\vec{V}_{21} \right | = m_2 \left |\vec{V}_{22} \right |-m_1 \left |\vec{V}_{12} \right |\end{align}

Defining $ m_1 $, for example, to be equal to one unit of mass, it is possible to calculate $ m_2 $ (measuring the values of the velocities of the particles before and after the collision is a standard procedure that can be carried out).

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what's wrong in considering the conservation of linear momentum as a consequence of Newton Laws? If you assume $F= \dot p$, which is true in classical mechanics, 3rd law and conservation of linear momentum are completely equivalent. –  pppqqq Jul 8 '13 at 16:36
Conservation of linear momentum is fundamentally the consequence of space translation symmetry, Newton's first and second laws are special cases. –  Constantine Jul 8 '13 at 17:18

protected by Qmechanic Jul 7 '13 at 17:30

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