1st law of Newton as a special case of the 2nd law (historical aspect)

• I
• MichPod
In summary, the first law sets the stage for the rest. It's utmost important to understand that the 1st law is a prerequisite to be able to state the 2nd law.
MichPod
I am speaking here not of the modern definitions (like a 1st Newton law as a definition of an inertial reference frame), but rather on the way the 1st law was formulated in the time of Newton.

That is, it's obvious that the 1st Newton law in its original formulation is a corollary of the 2nd Newton law, specifically for the case when the net force is zero, the acceleration is zero (F=ma), therefore the body would have for such a case a constant velocity or be at rest (the later, again, is a special case of the constant velocity).

Newton, obviously, was a genius, then the question should be raised why he formulated the 1st law as a separate law (he could not miss that one is the corollary to the other) and is there any additional value/content per the 1st law as per what it was seen in the times of Newton?

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Delta2
MichPod said:
why he formulated the 1st law as a separate law
Pedagogy, perhaps. When writing for an audience, you want to get the audience on your side. Start them off with something easy. Generate agreement. Establish a shared context. Then hit them with the interesting stuff.

In some sense, that's exactly the modern interpretation. The first law sets the stage for the rest.

Delta2
It's utmost important to understand that the 1st law is a prerequisite to be able to state the 2nd law. In modern terms the 1st law makes an assumption about the kinematics of point particles. In modern terms I'd formulate it as: "There exists an inertial frame of reference." For Newton there was even an absolute frame of reference, but that doesn't hold in view of what we know today. The paradigm today is that with the existence of one inertial frame there is an entire class of inertial frames, which cannot in any way be distinguished from each other. So a more pedantic formulation of the 1st law is: "There exists the class of inertial frames of reference." The operational realization of a frame of reference was most clearly worked out by Lang in the late 19th century. There are long debates about this question in this forum about this.

Only after the concept of inertial frames is established you can formulate the 2nd law which is one of the dynamical laws. It introduces the concepts of mass and force, and it's a topic of its own to make these definitions logically consistent.

Finally, together with the 3rd law, and thands to 3-4 centuries of theorizing today, I'd use symmetry principles making use of the action principle (Noether) to define Newtonian mechanics as the physics consistent with the Gaileo-Newton spacetime model with its 10-parametric Lie symmetry group as the most clear theoretical foundation. The operational definitions of the key fundamental quantities are still challenging but simpler than in the historical approach.

MichPod said:
I am speaking here not of the modern definitions (like a 1st Newton law as a definition of an inertial reference frame), but rather on the way the 1st law was formulated in the time of Newton.

That is, it's obvious that the 1st Newton law in its original formulation is a corollary of the 2nd Newton law, specifically for the case when the net force is zero, the acceleration is zero (F=ma), therefore the body would have for such a case a constant velocity or be at rest (the later, again, is a special case of the constant velocity).

Newton, obviously, was a genius, then the question should be raised why he formulated the 1st law as a separate law (he could not miss that one is the corollary to the other) and is there any additional value/content per the 1st law as per what it was seen in the times of Newton?
I've always assumed he needed to make a simple statement to break with the Aristotelian beliefs

Also, in the original paper he explains with some examples what the first law means. E.g. a spinning top only slows down because of the retarding action of the air. Whereas, planets free from such retarding forces may continue in their orbits for a much greater time.

He was taking things one step at a time.

Klystron, MichPod, jbriggs444 and 1 other person
vanhees71 said:
Only after the concept of inertial frames is established you can formulate the 2nd law which is one of the dynamical laws. It introduces the concepts of mass and force, and it's a topic of its own to make these definitions logically consistent.
The modern view of the Newton laws is exactly what I wanted to avoid in this discussion. Today we, arguably, may redefine and reinterpret Newton's laws and still call them Newton's.
But my point was that in the time of Newton and according to his original definitions and views, the 1st law must have been only(!?) a corollary of the 2nd one, not an establishment of the inertial frame reference notion and even(!?) not an establishment of the absolute frame of reference.

vanhees71
Well, may be. I'm not an expert on the Principia, which is pretty hard to read nowadays. One has to learn a lot of good old Euclidean geometry in its non-analytic form to really understand the details of the Principia. What we call "Newtonian mechanics" is due to Euler. The same holds for Maxwell's electrodynamics. To read the treatise I also have a hard time, needing some quartenion calculus. What we call Maxwell's theory today is due to Heaviside et al who translated it to the more modern vector calculus.

Klystron and Delta2
vanhees71 said:
Well, may be. I'm not an expert on the Principia, which is pretty hard to read nowadays.
I am not competent in this topic either. :-)
But we all know that there exists an old textbook formulation of the 1st Newton law which does not talk on the inertial reference frames nor of reference frames at all. Interestingly, my high school textbook was quite a hardcode, giving a modern formulation of the 1st Newton law, while my university textbook gave the old formulation along "a body will remain at rest or in a state of moving at a constant speed in a straight line unless it is acted upon by other bodies", then only mentioning that "the 1st law is not true in all reference frames"

So I generally assume that the later definition is somehow related to the old original definition of the Newton's time.

MichPod said:
The modern view of the Newton laws is exactly what I wanted to avoid in this discussion. Today we, arguably, may redefine and reinterpret Newton's laws and still call them Newton's.
But that is not what scholars of science do! They don't attribute things to scientists unless they can present valid arguments that those scientists are deserving of those attributions. What Newton's 1st Law tells us, in no uncertain terms, is that a state of rest is equivalent to a state of uniform motion. Nowadays that notion has been generalized to the Principle of Relativity, which is the assertion that all inertial reference frames are equivalent. But Newton's argument, that a state of rest is equivalent to a state of uniform motion, is essentially the same thing. And the idea had already been expounded in great detail by Galileo. Newton used it with great effect.

MichPod said:
But we all know that there exists an old textbook formulation of the 1st Newton law which does not talk on the inertial reference frames nor of reference frames at all.
That's only because that vocabulary didn't exist at the time. But the notion did.

Interestingly, my high school textbook was quite a hardcode, giving a modern formulation of the 1st Newton law, while my university textbook gave the old formulation along "a body will remain at rest or in a state of moving at a constant speed in a straight line unless it is acted upon by other bodies", then only mentioning that "the 1st law is not true in all reference frames"

So I generally assume that the later definition is somehow related to the old original definition of the Newton's time.

There are many college- and university-level introductory physics textbooks that state the 1st Law is a consequence of the 2nd Law. But those claims have all been debunked in the literature.

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MichPod said:
I am not competent in this topic either. :-)
But we all know that there exists an old textbook formulation of the 1st Newton law which does not talk on the inertial reference frames nor of reference frames at all. Interestingly, my high school textbook was quite a hardcode, giving a modern formulation of the 1st Newton law, while my university textbook gave the old formulation along "a body will remain at rest or in a state of moving at a constant speed in a straight line unless it is acted upon by other bodies", then only mentioning that "the 1st law is not true in all reference frames"

So I generally assume that the later definition is somehow related to the old original definition of the Newton's time.
Sure, there are many textbooks missing the point. Lex I holds only in inertial reference frames. It's an operational definition for establishing an inertial reference frame. Without much discussing it in the intro lecture we simply use the restframe of the Earth as an inertial frame, taking into account the gravitational force of the Earth on the bodies we describe explicitly as a force. Only a bit later we mention that this is in fact an approximation and then we demonstrate the Foucault pendulum :-)).

Mister T said:
But that is not what scholars of science do! They don't attribute things to scientists unless they can present valid arguments that those scientists are deserving of those attributions.
Well well, I am not pretending to be even an amateur historian of physics. I did not study much the sources before raising this topic, mainly because I thought there is an agreement that the old formulation of the 1st Newton law sounded like that 1st law is a consequence of the second one. And I would always like to be corrected.

Mister T said:
There are many college- and university-level introductory physics textbooks that state the 1st Law is a consequence of the 2nd Law. But those claims have all been debunked in the literature.
I guess that the statement in these textbooks is not merely an error of the authors of each textbook, but rather some old common understanding of the 1st Newton law, may be steaming from the Newton's work themselves. It's interesting to understand whether it was an original "fault" of Newton or some later misinterpretation of some authoritative source.

PeroK and weirdoguy
Well, many introductory textbooks are not very careful in stating Newton's postulates and usually they don't even discuss the important point that Newtonian mechanics is formulated in the class of inertial frames. The equations of motion in terms of coordinates defined wrt. non-inertial frames are still this very same laws only expressed in such non-inertial coordinates. This usually is not made clear enough in introductory textbooks and that's where the confusion illustrated nicely in this thread comes from. I think Newton's idea are much more profound than what's said in such textbooks, though of course he was wrong in his assumption of an absolute inertial frame of reference.

1. What is the 1st law of Newton as a special case of the 2nd law?

The 1st law of Newton, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will remain in motion at a constant velocity unless acted upon by an external force. This can be seen as a special case of the 2nd law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

2. How did Newton come up with the 1st law as a special case of the 2nd law?

Isaac Newton developed the 1st law of motion as a special case of his 2nd law through his extensive experiments and observations on the motion of objects. He observed that objects tend to maintain their state of motion unless acted upon by an external force, which led him to formulate the law of inertia.

3. What is the historical significance of the 1st law as a special case of the 2nd law?

The 1st law of motion as a special case of the 2nd law is historically significant as it marked a major shift in the understanding of motion and mechanics. It challenged the Aristotelian view that objects required a continuous force to keep them in motion and paved the way for the development of modern physics.

4. Can the 1st law of Newton be applied to all objects?

Yes, the 1st law of Newton can be applied to all objects, regardless of their size, shape, or composition. It is a fundamental law of physics that governs the behavior of all objects in motion, from microscopic particles to massive celestial bodies.

5. How does the 1st law of Newton as a special case of the 2nd law relate to everyday life?

The 1st law of Newton as a special case of the 2nd law has many practical applications in everyday life. It explains why objects tend to stay at rest or in motion unless acted upon by a force, and helps us understand the concept of inertia. It is also the basis for many inventions and technologies, such as seatbelts, airbags, and roller coasters.

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