I Extending Newton's laws -- Is the concept of force still defined?

[hutchphd]

I won't stop you.

 

[Orodruin]

I would propose to restrict the definition of "body" to baryonic matter (using it in the general sense to include electrons in atoms). After all, the concept of force originated with macroscopic objects containing baryonic matter. I realize that this may not work if you want to include as bodies such objects as dark matter, neutrinos, free electrons, black holes etc. However, the concept of force is not particularly useful, if it can apply at all, to such objects.

How about: "A body is an object comprised of baryonic matter, the quantity and identity of which does not change with time".

Baryonic matter appears in relativistic quantum field theories. It has very little to do with classical forces and classical forces are not very useful in the Standard Model either. Already with baryonic matter you run into issues of mass non-conservation.

We could then define "force" as a physical interaction with the body that changes the motion of a body.

If we accept your definition of "body" as baryonic matter with fixed constituents, you are already out of the constant mass case - as required by relativistic quantum theory.

Either way, your post is full of "we could define" and similar. Even if there were no caveats in the definition of what constitutes a "body", you could also do it in different ways that all capture the essence of Newton's laws of motion when applied to the constant mass case.

 

[Dale]

If by ”force” you mean ”any momentum exchange” (including momentum lost or gained by mass entering or exiting your system) then Newton’s second law F = dp/dt holds as is.

As a matter of personal preference I would define force as the rate of change of momentum with respect to time. That makes it easy to generalize to relativity where the rate of change of the four momentum with respect to proper time is a tensor.

 

[Andrew Mason]

As a matter of personal preference I would define force as the rate of change of momentum with respect to time. That makes it easy to generalize to relativity where the rate of change of the four momentum with respect to proper time is a tensor.

It is certainly the case that force=dp/dt. But if one defines force as dp/dt then one has to define momentum. If momentum is defined as mass x velocity, one has to define mass in a quantifiable way. But the only way to define mass in a quantifiable way (without resorting to something based on the number of protons+electrons + neutrons) is by its inertia ie.: m=(dp/dt)/(dv/dt) and everything becomes circular. I think the OP was trying to avoid that.

AM

 

[Dale]

If momentum is defined as mass x velocity,

Then momentum can be defined through Noether’s theorem.

 

[strangerep]

Let me try to pose post 1 in another form then. Hypotheses once more: two bodies of masses $m_1 (t)$ variable which is acted on by a body of mass $m_2$ (constant) through a force we call it $\vec{F}_{2,1} (t)$. The effect of the body $m_2$ is to make the other one both move and lose mass in time. Question: can we use the 3 known Newton's laws to calculate their motion ($\vec{x}_{1} (t), \vec{x}_{2} (t)$)? If yes, how? If not, why?

As others have said, this scenario is incompletely specified. Where does the lost mass go? Can you formulate a Lagrangian that covers all entities in the system?

[Btw, I symphathize with your frustration about how your thread keeps being subtly hijacked.]