Does a point mass have constant mass?

Expanding on the title, I am hoping for a little bit of insight into the philosophy of Newton's mechanics as a physical theory. My question, essentially, is does a point mass have constant mass by definition?

Elementary textbooks (I am using Taylor's Classical Mechanics) introduce Newton's three laws in the context of point masses. Thus am I to take this as the tacit suggestion that those three laws are taken as axioms which apply to point masses only, wherein a point mass is defined as a constant mass with only translational DOF?

Edit: Newton's mechanics seem surprisingly rife with philosophical quandaries. If anyone has a suggested resource explicating these I would also greatly appreciate it.

• "My question, essentially, is does a point mass have constant mass by definition?" If you define it as such, then yes. How does your textbook define "point mass?"
– hft
Commented Mar 22, 2023 at 2:41
• @hft It does not make precise what's meant by point mass. I'm therefore asking for the general set of axioms relating to point masses and Newton's 3 laws.
– EE18
Commented Mar 22, 2023 at 2:44
• OK, Here: Axiom 1: A point mass by definition has a constant mass. This is reasonable since, in practice, the point masses we know of in nature indeed have constant (rest) masses.
– hft
Commented Mar 22, 2023 at 2:46
• Not sure what other kind of answer you are looking for, but maybe someone else will be able to respond/answer.
– hft
Commented Mar 22, 2023 at 2:47
• Newtonian mechanics biggest quandary is that it is wrong, but right enough and still very useful
– JEB
Commented Mar 22, 2023 at 4:51

Yes. A point mass is constant by definition. For example, planets are considered point masses in orbital mechanics and these masses are assumed constant.

Another way to think of a point mass is that it's an object's maximum theoretical density.

• Noted. Further, do we consider Newton's 3 laws as (axiomatically) only applying to point particles (with corresponding results for extended bodies as derived theorems)?
– EE18
Commented Mar 22, 2023 at 3:54
• Would it be possible to comment on this question? I would be happy to accept if you can. Thank you!
– EE18
Commented Sep 4, 2023 at 20:51
• Turns out there is a SE post that addresses your question: physics.stackexchange.com/q/492560/334569 Commented Sep 5, 2023 at 0:26
• I am not sure that any answer there is completely authoritative unfortunately :( I would love a nice list of the classical mechanics axioms, stated rigorously.
– EE18
Commented Sep 5, 2023 at 0:40
• Im not sure I can provide what you are after. Newton's Laws most certainly apply to point particles as an axiom. But they also apply to collections of distributed point particles with complex interactions. The best example I can give is how Newtons laws apply to rockets where the thrust of the rocket is provided by expelling rocket mass in the form of fuel. Commented Sep 5, 2023 at 1:16

This question points to the definition of a point mass. As usual with definitions, they are conventional, i.e. they are rooted in the phenomena but depend on the particular theory they are embedded in, and people have to agree about their usefulness.

In the case of Newtonian mechanics, at the macroscopic level, we see that only extended objects exist. The concept of a point particle is just a convenient approximation, valid as far as the size and the shape of the body do not play any role in determining its dynamics.

In the case of extended bodies, Newtonian dynamics was initially introduced to describe the motion of fixed-mass bodies, and mass was introduced as a fixed property of each body. However, the theory can be easily extended to deal with systems of bodies and then to treat also the case of varying mass bodies, i.e., bodies that lose mass by ejecting or adding other physical objects having mass (gas, powder, pieces of the body, external bodies, ...). How to deal with such cases of varying mass is a well-known chapter of Newtonian mechanics, deducible from the case of fixed-mass, and not reducible, in general, to the simple equation $$\frac{{\mathrm d}\vec p}{{\mathrm d}t}=m \frac{{\mathrm d}\vec v}{{\mathrm d}t}+\vec v \frac{m}{{\mathrm d}t}$$. But this is a different story.

Relevant to the present question is the observation that if the point mass approximation is introduced to neglect the effect of the size and shape of a body, nothing prevents speaking about a varying mass point particle. The precise meaning of such an approximation is that it is helpful if we can adequately describe the movement by only using the equations of motion for a representative point of the body.

Elementary textbooks (I am using Taylor's Classical Mechanics) introduce Newton's three laws in the context of point masses. Thus am I to take this as the tacit suggestion that those three laws are taken as axioms which apply to point masses only, wherein a point mass is defined as a constant mass with only translational DOF?

The Newton's 3 laws are:

1. A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.(Law of inertia)
2. When a body is acted upon by a force, the time rate of change of its momentum equals the force.(effect of force)
3. If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.(collision and equilibrium)

In these laws the mass is not there it comes indirectly through the 2nd law via momentum $$\vec p = m\vec v$$.

Expressing Newton's 2nd law in mathematical form $$\vec F=d\vec{p}/dt=\vec v dm/dt + md\vec v/dt$$ You see even if the external force on a body is $$\vec F =0$$. There's still a way to accelerate that body if it is radiating mass.

That is the reason unwinding chains fall faster.*

Point mass is a hypothetical mass which is taken to be at a single point and in principle if you take it to radiating the Newton's law are still applicable. So there's no need to consider them constant.

But they are a construct to ignore all other effects of a rigid body like rotation etc.

• 1st law that body's momentum doesn't change arbitrarily. 2nd law how does momentum change, 3rd law what happens in two body collision Commented Mar 22, 2023 at 4:39