Why is the First Law of Motion a physical law?

Question

I'm learning physics on my own and this question I searched for answers but couldn't really find any discussions of this math/physics distinction.

The First law of motion is essentially

if $$\tag 1 \frac {d\vec v}{dt} = 0$$ then $\vec v=\text{const}$

there's little more stuff to it but if you look at the essence of the law then (1) is it.

This is a basic law from calculus, not physics. It seems like a physical law because it's talking about motion but rather (1) is the meaningful information here.

I think what's really going on is this:

(2) physical motions correspond to certain mathematical functions, something like that.

(3) Then mathematics, including (1) can be used to describe all motions.

We can call (2) the Law of Motion, then just use math freely. I don't think we need to specify something like (1) as any law of physics.

So once you have (2) properly configured and defined as the Law of Motion I don't think you need the First Law of Motion.

Answer 1

This is a basic law from calculus, not physics.

When Newton proposed the first law there were not any basic laws from calculus. Calculus was completely novel and unknown.

The first law was written because it contradicted the then-current theory of Aristotle which held that force-free motion was rest, not motion in a straight line at constant speed.

The First law of motion is essentially

(1) if dv/dt = 0 then v=constant

Almost. It is if $\vec F=0$ then $\vec v= \text{const}.$. Again, in contradiction of Aristotle’s if $\vec F = 0$ then $\vec v = 0$.

Answer 2

Newton's first law is not

$\frac {dv}{dt} = 0 \Rightarrow v=\rm{const}$

but

$F = 0 \Rightarrow v=\rm{const}$

In terms of modern mathematics, this would imply to us that acceleration dv/dt is some general but undefined function of force:

$$a=g(F)$$

without a constant additive term. But this is consistent with the force law $a=kF^2$, $a=k\sqrt F$, $a=kF^b$, or any number of others.

It takes the second law to postulate that the relationship is simple proportionality, and that the proportionality constant is the body's mass:

$$F=ma$$

Although in hindsight we may view the first law as a trivial case of the second law, the first law is a more general statement that can remain true even if the second law were falsified. And as others have pointed out, the first law in itself was revolutionary, as it contradicted the view of Aristotle that a constant input of effort was needed to maintain motion.

Answer 3

I very much disagree with your formulation of the first law. In fact, historically the first law was very important because it contradicted ancient wisdom passed down from the Ancient Greeks (as well as common sense!), that an object moving at a constant velocity now will eventually come to rest. This actually should seem quite sensible to you -- a rolling ball will eventually come to a stop, for example. The important insight of Galileo and Newton is that the fact that a rolling ball stops is not a fundamental aspect of motion, but a consequence of the existence of frictional forces (which, it turns out, far from being fundamental are actually very complicated to explain from first principles!).

Since your statement "If $\frac{dv}{dt}=0$ then $v={\rm const}$" is a mathematical triviality, it cannot be the content of the first law, because the first law contradicted a possible model of reality that turned out not to describe the real world when looked at experimentally in detail.

The way I would phrase the first law, in terms similar to yours, would be: "If a particle is moving with speed $v$ at time $t$, and no external forces act on it, then it will continue to be $v$ at all later times." You can infer from this that $\frac{dv}{dt}=0$, but we are not assuming $\frac{dv}{dt}=0$. If you want to derive $\frac{dv}{dt}=0$ from the statement "no external forces act on the particle", you need Newton's second law.

But more to the point, it is totally possible to consider a mathematical model where there are no external forces acting on a particle and $v$ is not constant. This is the model that Aristotle, et al, believed (well, they didn't really have anything as well defined as a mathematical model, but you certainly can build a model that has this feature). Newton's laws are important because they define a specific model -- which isn't the only theoretically possible model -- and this model was then confirmed by doing experiments. That is why the first law is a physical law, ultimately -- it is a non-trivial statement (ie, it doesn't follow deductively from any other assumption) that makes predictions about how the world should behave, and those predictions are confirmed by experiments.

Answer 4

Apart from the historical reasons, one thing that should be added to the excellent answers that already appeared is that, even in a modern perspective, where we can take for granted $F=ma$, the first law or some equivalent formulation is required as an implicit or explicit specification of the role of inertial reference frames. In a non-inertial reference frame, acceleration may be different from zero even in the absence of forces.

Answer 5

That's just not what the first law says.

"An object in motion will remain in motion unless acted upon by another force" means that if $\frac{dv}{dt}(t_o) = 0$ for any $t_o$, then it will be 0 for all t unless a force acts upon it.

This follows from the translational symmetry of space, as shown by Noether's theorem, it's not simply a consequence of calculus.

Answer 6

I would argue that the First Law simply forms the definition of inertial reference frames. In arbitrary reference frames it is certainly possible for objects to accelerate with no apparent force acting on them, with the resulting inclusion of "fictitious forces" necessary to describe their motion (such as the centrifugal or Coriolis forces).

So the First Law says that there exist certain privileged reference frames in which the deviation from linear motion of objects is entirely caused by external forces acting on them, whose sources have "real" physical existence (like springs, friction, electromagnetic forces, gravitational forces*, ...). An object on which no apparent force is acting will move in linear fashion, with a constant velocity, in inertial frames. So if you see an object that moves with some nonzero acceleration, and you have not been able to identify any "real" force acting on it, then you may deduce that you are in a non-inertial frame.

You could ask what exactly differentiates a "real" force from a "fictitious" force then. I would say that the distinction is also a matter of convenience/simplicity in the formulation of the laws of physics to fit experiment. In principle, we could have postulated that the reference frame of an observer on Earth is inertial. But upon looking at the sky, we observe all stars moving around the Earth, which would imply the existence of some strange force from the center of the Earth to all points in space with magnitude proportional to the distance. While such geocentric models are possible in principle, they are not very elegant, and a far simpler explanation of all these observations can be obtained by understanding the Earth as undergoing rotation around itself and around the Sun.

As for the Second Law, I would argue that it forms a quantitative definition of either mass or force, depending on the specific ordering of your experiments. You could imagine a situation where you are trying to characterize an unknown object by its motion under known forces (such as experiments with springs, where you have postulated that your springs are repeatably able to produce a certain unit of force, under a well-defined extension length). Then you could use such springs on objects with unknown masses to relate acceleration with the force applied, and you would observe experimentally that the relationship is linear, with the proportionality constant being defined as the "mass" of the object. Alternately, you could have started from known test objects, which you would define as having some unit of mass, and observe their accelerations in certain unknown force fields, such as electrostatic forces produced by a point charge. You could then find how the acceleration depends on the distance between the object and the charged particle, and deduce Coulomb's law for the electrostatic force.

You would probably also be interested in this question and some of its answers.

*I will avoid discussing general relativity in this setting, but in general relativity gravitational forces are also "fictitious".

Answer 7

I think of this way, why is the 1st law necessary? I mean what if it was not proposed. Then also we can derive it from the equation $F = m\frac{dv}{dt}$ by putting $F=0$. So where does the need for 1st law arise?

The answer is that to define an inertial frame of reference, the 1st law is necessary. But how? Imagine any object. You can always choose a frame of reference such that it appears at rest. And since $F$ in the equation is a real force (not pseudo force), nothing changes. But the 1st law states that, in an inertial frame of reference, if an object experiences net 0 force, then it continues with the same velocity. If you read this carefully, it defines what an inertial frame of reference is. It is also a test to determine whether an arbitrary frame is inertial or not.

Answer 8

It's a law because (a) It seems to always, accurately describe the natural behavior of something (actually, in this case, a lot of things,) and (b) because it can be concisely expressed in mathematical form (see answers by Dale and RC_23.)