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I'm looking to get rigorous definitions on which to base the important quantities in classical mechanics. To me a "rigorous" physical definition is an operational definition -- that is one in which we define how to measure the quantity over the course of an experiment -- or a definition in terms of other quantities which are operationally defined.

Time can be measured (at least in classical theory) arbitrarily well with clocks. Whether or not we can actually build a clock with the necessary accuracy is not important for this definition as long as we can conceive of one being built.

Position can be measured arbitrarily well with measuring sticks.

Using these two quantities above we could conceive of strategically placing an array of measuring sticks and clocks (that is building a reference frame from them) so as to entirely measure the motion -- position, velocity, acceleration, etc as a function of time -- of any object we care to.

Now I turn my attention to the most important equation in Newtonian mechanics: $F=ma$ (or $F=\dot p$ it doesn't really matter in this case). For this equation to make any physical sense we need to be able to measure the mass and/ or force (and/ or momentum) of an object during any experiment arbitrarily well.

How can we define force or mass (or momentum) in such a way that in the midst of some complicated motion -- possibly involving losing/gaining mass, etc -- we can still be certain that we can measure values for these quantities?

One note on this: I would prefer not to assume the equivalence principle as a way of measuring mass. Let's just pretend that inertia mass and gravitational mass are not the same because, as far as I can tell, in classical mechanics there is no strong reason to believe the equivalence principle holds based solely on Newton's laws (and other equally important "first principles" like the conservation laws). Of course, there's empirical evidence, but again let's just ignore this for now and see if we can find another way of measuring mass (or force or momentum or kinetic energy or any other thing which as allow us to obtain a measurement of mass indirectly).


EDIT: This will be a response to the question that Qmechanic links to (admittedly a similar question) detailing why I find the answers therein unsatisfactory. My hope is that either someone can give a new answer or can assuage my uneasiness about one or more of those answers to the linked question.

The top answer is by joshphysics.

The part about forces (and masses) is his statement of the third law: " If any two objects are being observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant." Here he implicitly assumes that two objects in an inertial frame will be accelerating. So I think he just forgot to add that there must also be so interaction (force) between the two objects.

If we assume that there is an interaction between the two objects then this is just Newton's regular old third law (pretty much). However, this definition only provides a means of measuring mass if the accelerations of the two bodies are completely due to the single interaction between the two objects.

I'm not entirely sure I understand his second law and so let's move on.

The second answer is by Cleonis.

His second law is the standard second law. Cleonis claims that he's defining force and mass but he doesn't (well he does seem to imply that given a known force that mass is just a constant of proportionality in the last sentence of his third-to-last paragraph before the additional remarks -- but without a way of measuring force, it's not useful) -- so that's a bit disappointing.

Constantine then defines force as the time derivative of momentum.

Even if we leave force as a primitive we still need a method of measuring the mass of an object in any given system. His method of defining it in a system of two particles doesn't work if we have more than two particles so it has a very limited applicability.

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    $\begingroup$ The rigorous definition of force is the curvature of a fiber bundle. For the standard model forces we look at the principal bundle of $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ and for gravity we look at the tangent bundle. However, I suspect this is not the answer you are looking for. $\endgroup$ – Ryan Unger Apr 3 '15 at 1:11
  • $\begingroup$ Yeah... Is that the standard model as in QFT (I honestly can't tell because that's way over my head)? Because I'm just trying to get well defined concepts for understanding classical mechanics? $\endgroup$ – Bob Dylan Apr 3 '15 at 1:14
  • $\begingroup$ Yes, $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is the gauge group of the standard model, which is just an elaborate Yang-Mills theory. I know you are talking about classical mechanics, hence why this is a comment. Just some trivia for you to look forward to down the road. $\endgroup$ – Ryan Unger Apr 3 '15 at 1:17
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/70186/2451 and links therein. $\endgroup$ – Qmechanic Apr 3 '15 at 1:32
  • $\begingroup$ Related: How to derive or justify the expressions of momentum operator and energy operator? (PSE/q/83902). $\qquad$ p.s. Bob Dylan: "Position can be measured arbitrarily well with measuring sticks." -- $\qquad$ No: Distance of pairs of participants ("ends") can be measured (trial by trial); and if so, the applicable pair in the applicable trial is called "(ends of) a measuring stick". (Likewise: Duration can be measured ...) $\endgroup$ – user12262 Apr 4 '15 at 4:41
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Can we rigorously define force?

Presumably this question is asked in the context of Newton's mechanics (not to be confused with Newtonian mechanics). I'll be unconventional and say the answer is "no".

Newton's three laws and his first few corollaries do not define force. They instead describe what forces do and how they relate to mass (also undefined), how they relate to one another, and how they behave mathematically. Newton's definitions (which precede his laws of motion) attempts to define concepts such as time, mass, velocity, and force, but these are rather vague.

To me, a better approach would be to make force, time, and mass "undefined terms". For the best example of this, look to Euclidean geometry. What are the Euclidean definitions of a point, a line, and a plane? The answer is that there is none. These are the three undefined terms in Euclidean geometry. Just because those terms are undefined does not mean they are useless, or that one cannot talk about relationships between them.

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Sure we can rigorously define force: It's a number extrinsic to a mass that allows us to calculate the mass's acceleration given also the numbers assigned to its intrinsic physical properties.

We only require the measurement of force to give us a number that correctly predicts the measured acceleration of the mass. If it does this, then the measurement is defined correctly.

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I hope not to be boring with the following sentences. There are two more practicable characteristics for masses and velocities. First is the momentum $p = mv$, second is the energy $E = 1/2 m v^2$. Both are conserved values for a closed system and for both one has not to care about the relative speed of an observer (with the exception that we talk about non relativistic speed and not taking in account gravitational influence). One get two equations for two interacting bodies with masses $m_x$ and velocities $v_x$ with $p_b = p_a$ and $E_b = E_a$ (b means before and a means after the interaction). Of course there are forces between the two bodies but we have not care about them.

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  • $\begingroup$ My question is really about operational definitions. So, if we use $p=mv$ and $E=\frac 12 mv^2$, to find $m$, we'd need a way to either measure $p$ or $E$. Or do we have some way of identifying (and assigning a measurement to) the conserved quantities in a system just by looking at it? $\endgroup$ – Bob Dylan Apr 3 '15 at 13:00
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Well, newton's 2nd law states that force is directly proportionate to the change in momentum over time so a force ,genetically, is a momentum changer. But, the concept of time is also imbued in the term 'force' so momentum changes per unit time.

My own definition : Force is a manipulator of momentum. Every unit time, the momentum changes when it is subjected to force.

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