# Formalism of Newtonian Mechanics

#### loom91

Hi,

I was wondering, how would one formulate Newtonian Mechanics as a rigorous mathematical model? Would one take force to be an external quantity defined by various force laws (Coulomb, UG) and accept Newton's Second Law of Motion as an axiom (and first law as a definition of inertial frames)? Or would one take the first and second law to be definitons of force (in whitch case it seems to me the concept of inertial frame becomes impossible to define)? Also, would one take mass to be externally defined?

It seems to me both of the approaches, which are mathematically incompatiable, are used in textbooks. What's worse, they are intermixed according to convenience, confusing me about inertial reference frames.

While it seems that later formulations of classical mechanics (Lagrangian, Hamiltonian, Hamilton-Jacobi) are introduced in a rigorous manner, the structure of Newtonian Mechanics is never presented in any detail. I am a fanatic for rigour, proofs and highly abstract and formal statements (even though I prefer physics over mathematics any day). For me this state of ignorance is rather disturbing. Can you help me out?

Thanks.

Molu

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You don't have to be alarmed about that, since Newtonian mechanics in its 'most innocent' form is taught for a good reason before something more, as stated - rigorous, such as Lagrangian, Hamiltonian etc.

#### loom91

Do you mean it is not possible to give a rigorous foundation to newtonian mechanics?

#### loom91

Well? Should Newton's laws be treated as definitions or laws?

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loom91 said:
Well? Should Newton's laws be treated as definitions or laws?
There are definitions in mathematics, not in physics. At least not in this context. So, Newton's laws are laws. You are probably refering to classical mechanics in general - which means non-relativistic or non-quantum mechanics. Maybe it's best to say that Newtonian mechanics means classical mechanics at an earlier stage of development. Classical mechanics, including the Lagrangian and Hamiltonian approach, does not include a 'force' in such an explicit way as Newtonian mechanics does; it's more, let's say, based on an energetic approach, as far as I got it.

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#### ZapperZ

Staff Emeritus
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Why are we so concerned about giving labels to these things? This is as nonsensical as calling Pluto as planet or not. It make NO difference! Call it a monkey if it pleases you. It changes nothing.

Zz.

#### loom91

ZapperZ said:
Why are we so concerned about giving labels to these things? This is as nonsensical as calling Pluto as planet or not. It make NO difference! Call it a monkey if it pleases you. It changes nothing.

Zz.
Whether or not we call Pluto a planet is entirely a linguistical issue, but whether or not N2L is a definiton is not at all so. In any theory, one starts with certain definitions, certain axioms and certain allowed logical manipulations and then proceeds to derive theorems by applying these logical transformations on the axioms.

If N2L is a definition of what we call force, then we can not say that the centrifugal force is a non-existent or pseudoforce. It produces acceleration in one reference frame, therefore it must be a force. There can be no fundamental distinction between an inertial and a non-inertial frame. If however we take N2L to be a law dealing with forces defined via external force laws, then we can say centrifugal force is non-existent since there are no physical interactions that can give rise to this force.

It is not true that there are no definitions in physics.

#### ZapperZ

Staff Emeritus
2018 Award
OK, since you care that much about definitions, tell me what is a "law", and how is this different than a "theory"? Can a theory somehow "graduate" into a law? Where has this happened in physics?

There ARE definitions in physics, but these definitions are based on mathematical formalism, rather than callling it with labels such as "law", "axioms", etc. There is no ambiguity when you concentrate on what matters the most - the mathematical formalism. It becomes an exercise in futility like this if you only care about semantic labels!

If you are bored and have nothing to do, look this up: figure out the conservation laws for energy and momentum and why they are important via the Noether theorem. Then see why such conservation laws results in ALL of the classical mechanics that you are familiar with.

Zz.

#### loom91

ZapperZ said:
OK, since you care that much about definitions, tell me what is a "law", and how is this different than a "theory"? Can a theory somehow "graduate" into a law? Where has this happened in physics?

There ARE definitions in physics, but these definitions are based on mathematical formalism, rather than callling it with labels such as "law", "axioms", etc. There is no ambiguity when you concentrate on what matters the most - the mathematical formalism. It becomes an exercise in futility like this if you only care about semantic labels!

If you are bored and have nothing to do, look this up: figure out the conservation laws for energy and momentum and why they are important via the Noether theorem. Then see why such conservation laws results in ALL of the classical mechanics that you are familiar with.

Zz.
The mathematical formalism is what I was talking about, not linguistics. In mathematics there is a clear distinguition between definitions and axioms. In the formalism of newtonian mechanics do we take force to be defined via N2L or do we treat it as an already defined quantity that is related to the already defined concepts of motion via the N2L?

#### actionintegral

Hi Loom91,

You and I are of like mind. I am keenly interested in formalizing physics. I participate in this groups because they force me to sharpen my communication and reasoning skills.

You would like Bertrand Russells 'Foundations of Mathematics'. He discusses such logical issues as the tautological definition of force (F=MA) and the nearly religious fanaticism with which people square velocity (i.e. kinetic energy!)

Of course, I just got this thread banned to the philosophy channel, but I found a kindred spirit!

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actionintegral said:
...You would like Bertrand Russells 'Foundations of Mathematics'. He discusses such logical issues as the tautological definition of force (F=MA) and the nearly religious fanaticism with which people square velocity (i.e. kinetic energy!)
What religious fanaticism? There is a reason for which there is a 'squared velocity' term in the experssion for kinetic energy.

#### actionintegral

Yes, it eliminates time as a parameter. No argument from me. Sometimes that is useful. But in my personal opinion, I think a little too much is made of the concept.

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actionintegral said:
Yes, it eliminates time as a parameter. No argument from me. Sometimes that is useful. But in my personal opinion, I think a little too much is made of the concept.
How does it eliminate time?

#### actionintegral

Energy is always conserved. If you know the location of an object, you know the velocity of the object. But you don't know when.

ax = .5 v*v

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actionintegral said:
Energy is always conserved. If you know the location of an object, you know the velocity of the object. But you don't know when.

ax = .5 v*v
Velocity is time dependent.

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#### ZapperZ

Staff Emeritus
2018 Award
loom91 said:
The mathematical formalism is what I was talking about, not linguistics. In mathematics there is a clear distinguition between definitions and axioms. In the formalism of newtonian mechanics do we take force to be defined via N2L or do we treat it as an already defined quantity that is related to the already defined concepts of motion via the N2L?
Are you doing this as a HISTORY project, or do you actually want to know start-of-the-art knowledge on these things? For example, if I tell you that practically everything that you encounter in first year Intro Physics is nothing more than various manifestation of conservation of energy (or mass+energy) and conservation of momentum (both linear and angular), would you believe that? Would such a statement, which I have made before when I briefly taught undergrad physics, would cause you to realize that everything you see is a consequence of those two principles?

And would you be able to connect, via Noether's theorem, on why those two conservation laws are merely reflections on two different symmetry principles that we observe for the world we live in?

So yes, I have already given you more than enough to show you why Newton's laws, for example, are simply manifestation of more fundamental descriptions of the universe that we know. This directly addresses your very first post.

Zz.

#### actionintegral

Velocity is time dependent.
That's right. But in the Energy picture, you sacrifice that information and replace velocity as a function of time with velocity as a function of space.

#### samalkhaiat

loom91 said:
The mathematical formalism is what I was talking about, not linguistics. In mathematics there is a clear distinguition between definitions and axioms. In the formalism of newtonian mechanics do we take force to be defined via N2L or do we treat it as an already defined quantity that is related to the already defined concepts of motion via the N2L
Physical theories (Newton mecanics included) are self-consistent, but logically not closed.
We have to distinguish between the pure mathematical reasoning and the usually employed physical reasoning. Mathematical reasoning consists of definite logical rules on adopted system of definitions and axioms. The process of mathematical deduction contains no furthur information other than those in the initial system of definitions and axioms. Mathematical assertions are valid for the abstract objects introduced by means of the definitions. However, if a correspondence has been established between these objects and real (physical) objects such that a mathematical structure, under certain conditions, gives a correct description of the behaviour of physical objects, we can say that we have found a physical realization of the mathematical theory.In other words, a mathematical theory becomes physical if a physical realization of its basic concepts has been found.
To construct a physical theory by deductive manner we must:
1) form a general set of concepts whose introduction is suggested by physical phenonena.
2) limit the range of application of these concepts by some sort of "fundamental principles" (here clever people show up)
3) show that the limited concepts, together with the mathematical relations between them, form a self-consistent scheme.
By observing nature, an ambiguous "interaction" between man and outside world takes place. Mysteriously this process enables us to form a system of "qualitative" concepts which are independent of our intervention. In many cases it is possible to provide our concepts with a "quantitative" features which bear the signs of the different procedures of experimental study of physical objects. Thus, from mathematical point of view, we must expect that our concepts are illdefined.

regards

sam

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality"

E Einstein

Geometry and Experience

#### homology

Try V.I. Arnold's book on Mathematical Methods of Classical Mechanics.

Although I have ask, you mentioned that the Lagrangian version seemed more rigorous. Why is this?

You should also consider what ZapperZ is saying (if somewhat without tact). Mechanics is largely the result of symmetries.

However, if you like you devle into logic and model theory and all kinds of stuff I've only read about that gets into the nitty gritty aspects of what makes a theory and why the theory of classical mechanics is fundamentally different than the theory of quantum mechanics, and from a logical point of view not merely the obvious physical differences.

Cheers,

Kevin

#### loom91

ZapperZ said:
Are you doing this as a HISTORY project, or do you actually want to know start-of-the-art knowledge on these things? For example, if I tell you that practically everything that you encounter in first year Intro Physics is nothing more than various manifestation of conservation of energy (or mass+energy) and conservation of momentum (both linear and angular), would you believe that? Would such a statement, which I have made before when I briefly taught undergrad physics, would cause you to realize that everything you see is a consequence of those two principles?

And would you be able to connect, via Noether's theorem, on why those two conservation laws are merely reflections on two different symmetry principles that we observe for the world we live in?

So yes, I have already given you more than enough to show you why Newton's laws, for example, are simply manifestation of more fundamental descriptions of the universe that we know. This directly addresses your very first post.

Zz.
I know all of that and am quite aware of Noether's theorem and we have to prove Newton's laws from conservation principles, but this still does not answer my question: Are Newton's laws definitions or postulates?

#### samalkhaiat

loom91 said:
Are Newton's laws definitions or postulates?
If you ask Newton, he would say: niether.
Newton believed in the empirical/inductive methods. According to him the laws are simple generalizations of experience. Such generalizations are statements in which the predicate "add" something to the subject. By this I mean that the merit of the law lies not just in its ability to explain observed phenomena but also to predict a new one.
However, if you ask Einstein the same question, he would (as a believer in the theoretical/inductive methods) say: Newton's laws are postulates. He would place them in No(2) of my post #18.
Answering your question does not need Noether theorem which is a statement about symmetries. This is because, you could ask the very same question this time about the status of symmetries in the structure of physical theories.

regards

sam

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#### Hurkyl

Staff Emeritus
Gold Member
loom91 said:
In any theory, one starts with certain definitions, certain axioms and certain allowed logical manipulations and then proceeds to derive theorems by applying these logical transformations on the axioms.
The mathematical formalism is what I was talking about, not linguistics. In mathematics there is a clear distinguition between definitions and axioms.
Actually, this is linguistics. The distinction is not as clear as you might think.

Mathematically, a theory is nothing more than a set of statements that is closed under implication. One can study a theory by writing down a list of "axioms" for it, but there is nothing inherently special about those statements. It's an arbitrary choice. For an analogy, consider the notion of studying a vector space by choosing a basis for it.

The difference between a definition and an axiom is purely syntactic. In practical terms, there is absolutely no difference between defining a term and leaving the term undefined, but writing an axiom for it. (In fact, we often call the latter an "axiomatic definition")

In fact, given enough background machinery (e.g. set theory), we never have to define anything axiomatically. For example, we could say "A group is anything satisfying these properties" rather than "These are the axioms of a group".

#### loom91

So do we say that force is anything that produces acceleration in a mass or do we say that a mass in unaccelerated unless a force acts upon it? To me this soes seem to make a difference because if we adopt the first viewpoint then I fail to see how we can define the concept of an inertial reference frame, used so often in mechanics. Doesn't the definition of inertial reference frame rest upon particles being accelerated in the absence of force?

#### samalkhaiat

loom91 said:
Doesn't the definition of inertial reference frame rest upon particles being accelerated in the absence of force?
NO. you do need to understand Newton's first law.

regards

sam

#### loom91

samalkhaiat said:
loom91 said:
NO. you do need to understand Newton's first law.

regards

sam
That is exactly what I've been trying to do from the beginning. Newton's first and second law as I know them states:

1)If no net external force acts on a particle, then it is possible to select a class of reference frames in which the particle moves with constant velocity. These reference frames are the inertial reference frames.
(Force externally defined quantity, a definition of inertial reference frames rather than a statement about forces)
+
When acceleration is measured from an inertial reference frame, F=kma(postulate about relation between three externally defined quantities)

OR

2)A particle moves with constant velocity unless forced to change this state of motion by the application of external forces.
(Qualitative definition of force as something that causes acceleration, no scope of introducing inertial reference frames)
+
F=kma(Quantitative definition of force)

The two set of statements are fundamentally different. In the second, N1L is not independent because it can be derived from N2L, while in the first N1L is independent because the statement of N2L needs it. Also note that the first set actually makes a concrete statement, while the second set is simply definition.

If one adopts the first set, the force laws (UG, Coulomb) can be treated as definitions of force and N2L an experimentally verifiable statement. If one adopts the second set, N2L becomes a definition and now the force laws become experimentally verifiable statements, and we see their experimental validity is only in a certain class of reference frames. One can then define these as the inertial reference frames, but this definition uses external force laws and does not follow from Newtonian mechanics itself.