# Is this a valid understanding of Newtonian mechanics?

This is a conceptual understanding of Newtonian mechanics. What the laws mean, how we know they're true, etc. I'm looking for criticism. I know this is really border line on the "don't ask questions that can't be answered" rule, but here we go anyway.

The Laws
First Law: A body retains its velocity unless acted upon by an outside force.
This first law is actually a definition, not an empirical statement. Body can be defined based on sense data, as can be velocity. But force is as yet undefined. As it is the only undefined term in the statement, the statement must be a definition. A force is defined as "that which is said to 'act upon' a body when that body's velocity changes, the immediate cause for a change of velocity". This contrasts with the Aristotelian definition, which is "that which is said to 'act upon' a body when that body has non-zero velocity, the immediate cause for a change in position".

Second Law: A body's acceleration is a function of its mass and the force acting upon it, according to the relation F = ma.
This second one is still just a definition. It's implied that mass is a function of the specific body. We can apply this to predict accelerations (see Applications, below).

Third Law: When one body exerts a force on another, that other exerts an equal and opposite force on the first. This is the only of the three laws that is empirical. It is not a definition as force was already defined by the first two laws, and it cannot be proven logically from the first two. It would have to be proven by some sort of experiment.

Empirical observations
The most glaring omission from the Laws is what can cause a force. Newton probably just meant it to be implied that forces were caused by collisions between bodies. In any case, you can demonstrate experimentally that collisions cause forces. This is the first apparent strength of Newtonian physics over Aristotelian physics. The Aristotelian definition of a force is valid (no such thing as a false definition), but in Newtonian mechanics it's much easier to express the relationship between collisions and forces.

Applications
We can apply these laws to calculate masses, forces, and accelerations.

1. Define a unit mass. Mass is an unchanging property of a body and therefore we may simply take an arbitrary body and define its mass as the unit.
2. Notice that when the same object collides with the same object in the same way, it has the same acceleration. For example, roll the same ball into it from the same height down the same slope, or hit it with a pendulum dropped from the same height. This provides inductive evidence that that sort of collision always produces the same force (F = ma, m is unchanging, a is unchanging, therefore F is unchanging).
3. Now apply that collision to other objects to deduce their mass. If an object accelerates x times as much as the object of unit mass, it has 1/x mass.
4. You can now measure mass.
• You can define force as the cause behind the compression of a spring. You can quantify it by the length of the compression. Jul 13, 2012 at 16:46
• Hi Jack, and welcome to Physics Stack Exchange! You're right that this more or less falls under the "questions that can't be answered" umbrella, but I think it could be improved into a good question. In particular, what do you think might be wrong with your understanding, and why? Is there a particular definition you're not sure about? etc. Jul 13, 2012 at 17:11
• I suppose what bothered me was the conception of the first two laws as mere definitions. In class they're presented as empirical facts. Jul 13, 2012 at 18:24
• What's the Aristotle definition of force? He doesn't give one as far as I can see. He uses the term without any precise definition, and without any sensible theory. Jul 13, 2012 at 20:13
• The only content of Newton's laws is that there is a conserved momentum, which is proportional to the velocity, and the proportionality constant is a property of the body. When momentum flows between bodies, we say a force is acting on the body. This is the concise statement of the laws, the precise original statement is just a historical accident. Jul 13, 2012 at 20:32

''Newton probably just meant it to be implied that forces were caused by collisions between bodies.''

This is definitely not what he meant. Gravitational forces were introduced by Newton although they don't act through collisions.

I'll freely concede that this is a personal opinion, but I think that to really appreciate Newtonian mechanics you need to understand that they arise from symmetries. They arise from conservation of energy and momentum, and Gaililean invariance. The conservation laws in turn derive from spacetime symmetries via Noether's theorem. There's a good discussion on the relationship of Newton's laws to the convservation of energy and momentum on Wikipedia.

If you're interested in getting a real understanding of mechanics you should also learn the Hamiltonian and Lagrangian approaches to mechanics as these have an elegance far beyond merely stating three laws.

• What is they in the first sentence? The equations of motion? And what does merely stating three laws mean, I guess you have to state some laws eighter way. And why is stating the equations of motion not elegant? You say they arise from the Noether theorem over the course of symmetries, but is there really an argument for what comes before what? You have to postulate some Lagrangian in both cases. In any case, I agree that the more recent ways of putting mechanics are pretty elegant. Jul 13, 2012 at 17:27
• the Newtonian formulation is underappreciated, imo: it comes with a rich geometric structure as well Jul 13, 2012 at 17:52
• @Christoph: It really doesn't. Newton's laws allow for any sort of forces, including forces that don't admit a symplectic structure. You can add an equal and opposite friction force between any pair of particles proportional to the square of the velocity difference for instance. To forbid this, you need that the force is conservative at least, and really, you need a concept of Lagrangian mechanics, which adds more to Newton. Jul 13, 2012 at 21:12
• @Ron: see mathbin.net/102296 ; what's the etiquette here - should I post this as an answer (which would be off-topic) or a new 'question' (which wouldn't really be appropriate either)? Jul 14, 2012 at 0:32
• @Christoph: You are saying simple things in a ridiculously annoying formal way. The condition $T\tau\odot F= id_\mathrm{TM}$ is obvious, it is just saying the force is attached to the right point on the manifold, it's obtuse because of the annoying fiber language. I don't see any point in this. Isn't the condition of exactness conservativity? Conservativity is not a consequence of Newton's laws. The formulation adds nothing as far as I can see, and is opaque, why are you making people use it? I think you're making people work a lot harder than they need to to follow you, for no gain. Jul 14, 2012 at 2:42

I don't see anything obviously wrong with your understanding, except for the fact that forces need not necessarily be mediated by collisions (Newton's law of gravity would be the prime example).

What is also interesting (at least I think so) is how these three laws translate to a geometric formulation on the cotangent bundle (momentum space).

The general structure of the theory is given by the second law: We need something called mass which maps tangent vectors (velocities) to cotangent vectors (momenta) and a force field of second order (basically the semi-spray condition).

The first law is a bit problematic: Force fields form an affine space, ie there's no natural way to define a force of zero on arbitrary base manifolds (configuration spaces): A zero point must be chosen explicitly. It turns out that such a choice is equivalent to the choice of a (covariant) connection, which plays a fundamental role in general relativity.

I'm actually not sure about the implications of the third law: The more general concept behind the third law is conservation of momentum, which - in the Lagrangian and Hamiltonian formulations - can be linked to translational invariance. In both cases, the force is derived from some sort of potential (the Lagrange and Hamilton function, respectively). I'll have to think about what this implies for the Newtonian formulation.

• Newton did not formulate the laws that apply to motion on a nontrivial manifold, as the cotangent bundle formulation does. The cotangent bundle formulation is fine, but it extends Newton's laws to Lagrangian mechanics, where the forces are conservative in the phase-space volume preserving sense. Jul 14, 2012 at 2:56
• @Ron: Newton might not have done it, but his first and second law tell us how to do it Jul 14, 2012 at 3:27
• In addition to the first and second law, you also need the fact that the forces that constrain the particle to the abstract manifold are conservative. This is not quite Newton's laws, it's Lagrange's laws, and the two are only equivalent if you assume the forces are conservative (and then there are cases which are Lagrangian and not strictly Newtonian conservative, like magnetic forces, which are velocity dependent). Jul 14, 2012 at 4:15
• @Ron: first of all, thanks for sticking with me so far - I hope your not just doing it to prove some know-it-all wrong ;) now, three arguments come to mind to counter your point: Jul 14, 2012 at 6:32
• (1) there are no constraint forces on spacetime, ie there's at least one genuine example where the above description must be considered fundamental (2) your objection holds true when deriving the EL equations from Hamilton's principle as well, ie having a formalism purely defined on some abstract space apparently doesn't bother most physicists (3) the derivation of the EL equations via d'Alembert's principle is a form of phase space reduction fully contained within the Newtonian framework and may of course impose additional restrictions on the forces Jul 14, 2012 at 6:32

The sentences you set in italic are probably ment to be qualitative explainations. You can do that, but I feel without using mathematics (and thereby taking a shortcut by implying the standard definitions in mathematics, e.g. regarding geometric quantities) the laws are very difficlult or impossible to put into a precise form.

Anyway, from the perspective that what you want is a clear prose, here are some remarks:

First Law

• Body can be defined based on sense data, as can be velocity.

Here you eighter want to have already an understanding what space, time and trajectories are and you identify these in some way with the notion of body (are you thinking of point particles or maybe something more complicated?). Otherwise I don't know what you mean by "define" here. If you take data, e.g. your shoesize, then you must already have an association what a shoes is. The shoe is what you want to define here in the sentece right? The sentence "something can defined based on sense data" seems confusing to me.

• But force is as yet undefined.

You use the word But here, implying that body and velocity have already been defined.

• This first law is actually a definition, not an empirical statement.

I saw this thread here recently discussing related topics. The threads referenced in the question might also be of interest.

• A force is defined as "that which is said to 'act upon' a body when that body's velocity changes, the immediate cause for a change of velocity".

Well, as the notion of absolute space is pretty much discarded today, there are some major complications relating to what I mentioned about math being a better suited language here, and some answer might also be found in the question I posted. Problem is that velocity is relative and you can't just take the definition you gave and leave it at that. There is a quote by Einstien going in this direction

"The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration." —Albert Einstein: The Meaning of Relativity, p. 58

Basically, if you drive around wildly on rollerblades on a street, then things that have constant velocity relative to you must also be moving zig-zag on the street, and if you take your explaintion literally, then these objects are forcefree. But only in the roller blades frame.

• First Law: A body retains its velocity unless acted upon by an outside force.

If you and your helmet are in free fall from a bridge, then the helmet will not move relative to you because all objects are said to be observed to fall equally. So here, by the statements posted by you alone, you would infer that no force is acting upon it.

Second Law

• This second one is still just a definition.

What is this here? The second law? If so, what is it a definition for? I don't think you mean the general statement "posting that law here is a definition of that law", which is tautological as it applies to all laws. You already gave a definition of force, acceleration is the second time derivative of position x (of the "body"). Maybe (but I don't really think that's the case) you mean it's a definition of mass, which you characterize as a body-dependend function (real number, I guess). In any case, there is the practical application of it all: The relation F=ma (really a differential equation F(x(t),x'(t),t)=m x''(t)) restricts the quantitiy x'(t) (or determines it, one the starting configuration is given). This also brings me to the point

• We can apply this to predict accelerations.

The acceleration is just F/m, and you have to find the force and the masses of particles before that and to get a mathematical expression for F you have to measure trajectories. So while it's true, I kinda feel the formulation "we can apply this to calculate trajectories." is more direct.

• Applictions

I'm not sure if I would dub the possibilities of calculating mass an "application" of the laws, or maybe only second to the calculation of trajectories. Becuase as far as I see, the only thing we want from mass is it's value to plug it back into another equation of motion.

As several other things, the phrase Mass is an unchanging property is also somewhat postulated. Maybe the application statur is more justified if you postulate inertial mass-graviational mass equivalence and Newtons universal law of graviation.

• inductive evidence

Is this a technical term, what does it mean?

Third Law

• It would have to be proven by some sort of experiment.

Using the word "proving" is considered a little dangerous in the natural sciences.

• This is the only of the three laws that is empirical. It is not a definition as force was already defined by the first two laws, and it cannot be proven logically from the first two.

I haven't wrapped my head around what the word empirical means or what it excludes, let me just remark that all the expressions of the forces, e.g. "F(x):=-C*x(t)" or "F(x):=-C/x(t)^2" have some empirical influence to them. Someone decided to plug them into the second law to compute the trajectory. (Or calculate a mass as you say ;)

What the laws mean,

It means what it says on the label. Laws of motion (of celestial objects). "Law" in the mathematical sense, i.e., just explaining what we see in the night sky. The "law" is not telling the planets how to move, but it is telling the mathematician how he should write his formulas to predict what he sees in the sky.

how we know they're true,

By carefully measuring, plain and simple. The same way we checked to see if Einsteins theories were true, or any other, for that matter. Newton's laws are only true up to a point, of course, being a simplification (just like any other theory). Which is no criticism at all, they are fine for what they were used for, and I'm pretty sure nobody back then could have dreamed up what we know about the universe these days.

I'm looking for criticism.

No criticism per se, just two pointers:

1) You are quoting the 3 laws of motion and seem to be mostly worried about definitions, implied meaning etc.. I did not read Newtons original publication, but I guess there were definitions of the words used in those laws. And if not, then he very likely implied that words like "mass", "body", "force" are used in their intuitive everyday sense (at that time). Back then scientists were certainly not deeply redefining reality all the time, like we are used to today (i.e, quantum mechanics, spacetime, field theories etc.). I am quite literally sure that he simply wanted to explain how stuff works, not impose a new "meaning" on life or the universe. All he is talking about is in plain sight (apples falling from trees).

If I recall correctly, he did not even have the concept of "momentum", but used "velocity" instead. Which is easy to explain, as "velocity" is directly observed, while "momentum" might be a bit less direct of an experience (though anyone having hit their thumb by a returning bowling ball will have a very good understanding of what "momentum" means :) ).

One can argue whether "gravity" was understood by anyone except Newton (or even him) back then, but it's no large stretch of the imagination. It's not that hard of a concept if you are only thinking macroscopically and in normal ranges of gravitation (i.e., no black holes, no big bang etc.).

2) I think you're overthinking a bit. At the time of Newton, science was in its infancy and much closer to philosophy. His theory is not so much a huge mathematical building as we are used to them today. Rather, he is deftly stating that simple things are simple, and sucessfully applying Occams Razor to idealised celestical problems (i.e., getting rid of any notions of "aether" or some weird fixed dark-sphere where the stars are glued on, whatever you had back then :) ). Thankfully, observation agreed with him.