On Newton’s First and Second Laws

People sometimes think Newton’s First Law of motion is implied by his Second Law, making the First Law superfluous. However, the two laws actually serve two very different purposes, and the first law is essential. Recall that the two laws are

Law 1:  Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

Law 2:  A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

The second law refers to “a change in motion”, but this wouldn’t tell us the motion itself without the first law. For example, consider a particle undergoing circular motion. Is this particle’s motion changing? Many philosophers and physicists from Aristotle to Galileo regarded circular motion as the most perfect and unchanging form of motion, and didn’t think it needed any active external intervention to maintain it. Applying Newton’s second Law (without the first) in that context would lead to very different conclusions. A fairly sophisticated level of abstraction was needed for scientists such as Descartes and Huygens (and, eventually, Galileo) to discern that an object would move at constant speed in a straight line if it were not subject to any external influences. There are no actual examples of purely inertial motion in our experience (because forces, including gravity, are everywhere), but this was a key insight leading to the scientific revolution, and today we take it for granted.

Even prior to stating his Laws of Motion, Newton had already expressed the basic idea of Law 1 in Definition 3:

Definition 3:  Inherent force of matter is the power of resisting by which every body, so far as it is able, perseveres in its state either of resting or of moving uniformly straight forward.

We can explain the crucial role of Law 1 by examining Newton’s use of it in the very first Proposition, proving Kepler’s area law for an orbiting body subject to a central force. He writes

Let the time be divided into equal parts, and in the first part of the time let a body by its inherent force describe the straight line AB. In the second part of the time, if nothing hindered it, this body would (by Law 1) go straight to c… but when the body comes to B, let a centripetal [impulse] force act and make the body deviate from the straight line Bc and proceed in the straight line BC…

He then explains how Law 2 (and its Corollaries) tell us the position of C, which relates the amount of force applied to the body to the amount of distance between c and C. Thus, Law 1 is needed to tell us what the unchanged state of motion consists of, which would put the body at c, and then from the applied force and Law 2 we can infer the position C resulting from the changed state of motion. We could not even begin to do this if we didn’t know what a state of motion is. In other words, it wouldn’t even be possible to apply Law 2 (it has no meaning) without Law 1 telling us what constitutes an unchanging state of motion. Law 2 tells us the difference between the unforced state and the forced state, but it doesn’t tell us what the unforced state is, so by itself it doesn’t tell us the forced state either. Something like Law 1 is essential.

Having said this, there are some important considerations to be mentioned. First, the phrase “uniformly straight” is undefined, and Newton himself famously commented that “geometry doesn’t teach us to draw straight lines, but requires them to be drawn”. In pure mathematics one can take an abstract axiomatic approach to defining things like “point” and “line”, but in physics the notions of “straight line” and “uniform speed” are ultimately derived from physical phenomena, things such as rays of light. Indeed, freely moving particles can be used to define straight lines and uniform speeds, so there is a certain circularity in the assertion that free particles move at uniform speed in straight lines. The usefulness of the inductive-deductive process is that large classes of phenomena share the same senses of “spatially straight” and “temporally uniform”, so these primitive concepts can be inferred from a small set of phenomena, and then applied to all phenomena. This is why the scientific method isn’t entirely circular or tautological. The principle of inertia serves as a highly successful organizing principle for our experience.

One reason modern readers often become confused is that they are naively taught Law 2 in the (non-relativistic) form F = m dx²/dt², which seems to imply that merely setting F=0 would yield Law 1, meaning a body with no external forces applied to it has no acceleration, so one may say it moves at uniform speed in a straight line. However, this only represents consistency, not redundancy. The stated equation, with no further specification, actually has no meaning, because we haven’t specified the coordinate system x,t. There are many systems of coordinates for which that equation is not valid. It is valid only in terms of a very special class of coordinate systems x,t called standard inertial coordinate systems, taken to represent rectilinear spatial coordinates and uniform time coordinates. Law 1 essentially specifies and defines these coordinates, by identifying “straight lines and uniform speed” with the motions of free bodies. Law 3 also plays an essential role in defining standard inertial coordinate systems by establishing the temporal foliation based on inertial synchronization. We stipulate that inertia is isotropic in terms of these special coordinate systems. For example, Law 3 implies that identical particles acting to repel each other from rest in a given frame will reach equal speeds and distances in equal times, so they can be used to inertially synchronize clocks.

As an aside, we note that, at least arguably, Newton’s form of Law 2 relates the applied force to changes in what he elsewhere called the “quantity of motion”, which isn’t just velocity v, it is what we call momentum, mv. The precise wording used by Newton is slightly ambiguous, complicated by the distinction between impulse forces and continuous forces, but on the usual reading, his Law 2 was F = d(mv)/dt, so the case F=0 doesn’t unambiguously map to dv/dt=0, because m might also change such that mv is constant while v is changing. Indeed, with special relativity we know that the relation F=ma is wrong, if m is considered constant. A particle subjected to a constant force does not exhibit constant acceleration in terms of any standard inertial coordinate system. Historians differ as to whether Newton’s phrase “change in motion” in his statement of Law 2 can be assumed to be referring to a change in the quantity of motion, or should be interpreted as just referring to kinematic acceleration, but most readers assume the former.

Of course, Newton never suggested that the “inherent force of matter” might depend on the state of motion of the body. He clearly believed that an elementary particle of matter subjected to a constant force would exhibit constant acceleration in terms of any standard system of inertial coordinates. He made this clear by proving that bodies move among themselves in exactly the same way, whether they be referred to what he called absolute rest or to a system moving at uniform speed in a straight line. In Book 3 of Principia, Newton famously identified the center of mass of the solar system as the absolute rest frame, but it’s overlooked that he said we could equally well use any frame that is moving uniformly in a straight line relative to this frame, and that the choice of the solar system’s center of mass was purely conventional. In his proof of Proposition 11 in Book 3 he says the center of mass of the solar system is either at rest or moves uniformly in a straight line, and he says we can hypothesize that “the center of the word” is immoveable (breaking his own rule against unsubstantiated hypotheses). His purpose was not to establish some metaphysical sense of absolute rest, but rather to refute the common view that either the Sun or the Earth were the center of the world. He was explaining that neither of those are the (inertial) center, but rather the center of mass of the solar system, which is indeed the closest approximation to a perfectly inertial system in our vicinity within the galaxy. Needless to say, Newton didn’t know about our galaxy, let alone other galaxies, so his focus on the solar system as “the system of the world” is understandable.

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