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I was thinking of some very basics concept when a doubt came to my mind, therefore I will briefly explain the argument that led me to the doubt so that the question will be clear.

Newton law states: $\vec F= m \ddot{\vec x}$

For what I know this isn't a definition of force but a relation between two different quantities. The force applied on one body is equal to the product of the body mass by its acceleration.

Gravitational law states: $\vec F_{12}= G \frac {m_1 m_2}{r^2} \hat x_{12}$

For what I know this also isn't a definition of force but is a relation between the force acting on a body (in the gravitational case) and some others quantities of the system.

At this point I thought that I don't know what is a force because I only know that is something that in general is equal to $m \ddot{\vec x}$ and that in the gravitational case is equal to $G \frac {m_1 m_2}{r^2} \hat x_{12}$. My idea now is that I have to consider the Gravitational law as the definition of force (in the gravitational case) and Newton law as the relation between two different quantities.

Qmechanic
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SimoBartz
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2 Answers2

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You are touching a quite delicate point at the foundation of Newtonian mechanics: the interplay between principles, definitions and basic objects the theory is built on.

Your doubts probably stem from the fact that different basic formulations of the basic principles have been proposed over more than three centuries, partially overlapping and ending up with a very unsatisfactory exposition in most of the textbooks. Unfortunately the issue is not just a problem of correct reconstruction of the historical development, but it touches directly what can be said or not about Newtonian forces.

I'll try to make short a long story.

From Newton up to the fist half of nineteen century, the dominant point of view was that forces and accelerations have different definitions and the Newton's second law was an empirical finding about their proportionality through a constant, the mass.

Around the middle of that century a different point of view was pushed forward by a movement, including Kirchoff, Mach and Hertz, which assumed the second principle as a definition of force. Moreover, force ceases to be a fundamental quantity the dynamics is based on, to become a nickname for $m \vec a$. It is quite clear that this is a completely different point of view and it is incompatible with the previous one.

More recently, there has been a resurrection of the Newtonian original approach, after a critic review of the main weakness of Mach's point of view.

At the best of my knowledge, no final assessment is ever appeared in the literature about the two main approaches. However, refined Mach-like approaches and neo-Newtonian approaches are frequently present in the best textbooks on Newtonian mechanics. What should be absolutely avoided is to mix them.

Quite schematically (many variations on the main themes exist)

in a Mach-like approach:

  1. one has to define and state the existence of an inertial reference frame without using the word force;
  2. mass is defined by analyzing collisions between point-like particles;
  3. force is defined as $m\vec a$;

in a neo-Newtonian approach:

  1. force is a primitive concept;
  2. inertial reference frames can be defined as the frames where any force-free motion is uniform and on a straight line;
  3. $\vec F =m\vec a$ becomes a principle, i.e. a statement collecting all the known experience, and $m$ is defined as the proportionality constant between force and acceleration (this is probably the point with the maximum number of variants);

Depending on the level of required conceptual rigor, these formulations could be considered satisfactory or not. Probably for practical purposes both can be used, but without mixing them.

An example of the practical relevance of a keeping a clean difference between the two points of view is the following.

How do we know that forces are invariant in every inertial reference frame? In the Machian approach, it is a trivial consequence of the invariance of the mass and of the acceleration. In the neo-Newtonian approach, either we have to add the requirement of such invariance to the concept of force, or we have to check it for each proposed force law.

GiorgioP-DoomsdayClockIsAt-90
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  • Do you happen to have any more information on the critique of Mach's approach? That seems interesting to me, given how Lagrangian mechanics avoids forces entirely. The existence of an alternative way of modeling motion which has no forces makes me intuitively like an approach which says forces aren't "special." – Cort Ammon Jan 15 '20 at 15:11
  • @CortAmmon-ReinstateMonica A well argumented critique of Mach's approach can be found in Bunge, M. (1966). Mach's critique of Newtonian mechanics. American Journal of Physics, 34(7), 585-596. – GiorgioP-DoomsdayClockIsAt-90 Jan 15 '20 at 22:32
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A force is simply a quantification of the external physical influences on a body. All forces can be modelled by $F=ma$ including gravitation.

Gravitation is simply a specific case of $F=ma$ where $a$ in this case is modelled by $a=\frac{Gm_2}{r^2}̂_{12}$ so the equations are in complete alignment with each other and $F=ma$ is an encompassing description for it.

Dominic Reeves
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