Why does Greiner call them theorems?

[Jiman]

The Newtonian1or classical mechanics is governed by three axioms, which are not inde-pendent of each other:
1. the law of inertia, 2. the fundamental equation of dynamics, 3. the interaction law, and as a supplement: the theorems on independence concerning the superposition of forces and of motions.

 

[Isaac0427]

The Newtonian1or classical mechanics is governed by three axioms, which are not inde-pendent of each other:
1. the law of inertia, 2. the fundamental equation of dynamics, 3. the interaction law, and as a supplement: the theorems on independence concerning the superposition of forces and of motions.

What exactly is the question here? And is there any more context to that quote (I assume you are quoting a textbook)?

 

[Jiman]

What exactly is the question here? And is there any more context to that quote (I assume you are quoting a textbook)?

"the theorems on independence concerning the superposition of forces and of motions."
Why don't Greiner(German physicist) call them the principal of superposition of forces and the principal of superposition of motions?
Why Greiner call them theroms?

 

[Drakkith]

Where are you reading this at/from? Is it from a book or an online article?

 

[Jiman]

Where are you reading this at/from? Is it from a book or an online article?

on page134

 

[Isaac0427]

"the theorems on independence concerning the superposition of forces and of motions."
Why don't Greiner(German physicist) call them the principal of superposition of forces and the principal of superposition of motions?
Why Greiner call them theroms?

This seems like semantics, but looking at the book, I think he is talking about the theorems regarding vector addition (i.e. components of vectors add independently). Under "lex quarta" he mentions that it is postulated that forces add as vectors, so I can only assume that he is referring to the theorems/properties of vector addition.

Regardless, he is saying that Newtonian mechanics relies on the principles of vector addition. It doesn't really matter what he labels them as.

 

[Jiman]

This seems like semantics, but looking at the book, I think he is talking about the theorems regarding vector addition (i.e. components of vectors add independently). Under "lex quarta" he mentions that it is postulated that forces add as vectors, so I can only assume that he is referring to the theorems/properties of vector addition.

Regardless, he is saying that Newtonian mechanics relies on the principles of vector addition. It doesn't really matter what he labels them as.

“the theorems on independence concerning the superposition of forces and of motions.”the principal of superposition of forces is a physical assumption
Is the principal of superposition of motions also a physical assumption?

 

[Jiman]

This seems like semantics, but looking at the book, I think he is talking about the theorems regarding vector addition (i.e. components of vectors add independently). Under "lex quarta" he mentions that it is postulated that forces add as vectors, so I can only assume that he is referring to the theorems/properties of vector addition.

Regardless, he is saying that Newtonian mechanics relies on the principles of vector addition. It doesn't really matter what he labels them as.

Is the principal of superposition of motions pure mathematical method?

 

[Ibix]

Under "lex quarta" he mentions that it is postulated that forces add as vectors, so I can only assume that he is referring to the theorems/properties of vector addition.

In fact, he explicitly uses the word "principles" in this paragraph.

Thereby the superposition principle of the actions of forces is postulated (principle of unperturbed superposition).

So I agree it's just semantics.

Side note: I've only read pages 134 and 135 of this book, and there are already two points I'd take issue with. First, Greiner's "Premises of Newtonian mechanics" #2 says that Newtonian mechanics uses a concept of absolute space. I don't think that's true. I think Newton did believe in an undetectable absolute rest frame, but to write that in 1989 without commenting that there's no useful difference between "there exists an undetectable absolute rest frame" and "there is no absolute rest frame" is very poor in my opinion. Second, immediately below the last of his four premises, Greiner comments that "the velocity independence of mass [is] lost in the special theory of relativity". That strongly suggests to me that he's building up to using relativistic mass, which is very much a deprecated concept. Most modern writers use "mass" to mean the invariant mass, and "total energy" to mean the relativistic mass - so be wary.

 

[Jiman]

In fact, he explicitly uses the word "principles" in this paragraph.

So I agree it's just semantics.

Side note: I've only read pages 134 and 135 of this book, and there are already two points I'd take issue with. First, Greiner's "Premises of Newtonian mechanics" #2 says that Newtonian mechanics uses a concept of absolute space. I don't think that's true. I think Newton did believe in an undetectable absolute rest frame, but to write that in 1989 without commenting that there's no useful difference between "there exists an undetectable absolute rest frame" and "there is no absolute rest frame" is very poor in my opinion. Second, immediately below the last of his four premises, Greiner comments that "the velocity independence of mass [is] lost in the special theory of relativity". That strongly suggests to me that he's building up to using relativistic mass, which is very much a deprecated concept. Most modern writers use "mass" to mean the invariant mass, and "total energy" to mean the relativistic mass - so be wary.

Could you recommend me a more authoritative textbook？Thank you very much!

 

[wrobel]

mechanics is governed by three axioms, which are not inde-pendent of each other:

not three, more axioms, and they are independent
V Arnold Math methods of classical mechanics

 

[Jiman]

In fact, he explicitly uses the word "principles" in this paragraph.

So I agree it's just semantics.

Side note: I've only read pages 134 and 135 of this book, and there are already two points I'd take issue with. First, Greiner's "Premises of Newtonian mechanics" #2 says that Newtonian mechanics uses a concept of absolute space. I don't think that's true. I think Newton did believe in an undetectable absolute rest frame, but to write that in 1989 without commenting that there's no useful difference between "there exists an undetectable absolute rest frame" and "there is no absolute rest frame" is very poor in my opinion. Second, immediately below the last of his four premises, Greiner comments that "the velocity independence of mass [is] lost in the special theory of relativity". That strongly suggests to me that he's building up to using relativistic mass, which is very much a deprecated concept. Most modern writers use "mass" to mean the invariant mass, and "total energy" to mean the relativistic mass - so be wary.

Is the principal of superposition of motions also a physical assumption?

 

[Jiman]

not three, more axioms, and they are independent
V Arnold Math methods of classical mechanics

Why are there so many mistakes in the book?

 

[vanhees71]

Mistakes in Arnold? It's likely to have typos in any book, but real mistakes in Arnold? I doubt it.

Concerning the question about "Newton's axioms", one should know the history of Greiner's textbooks. They are meant to be used by students who start the general theory course already in the 1st semester, which was a novum at German universities at the time the books are written. I remember too well, how disappointed I was when I started to study physics at the university that theoretical physics was supposed to start in the 3rd semester only. But that were the good old times before the socalled "Bologna reform" on European universities, so you were free to do whatever you want, you only had to pass the "Vordiplom" exams after 2 years and the "Diplom" exams at the end. How you got the knowledge to do so nobody cared. So I went in the 3rd-semester theory lectures as a freshman. Greiner's books were among the best sources to read about all the many things I couldn't understand lacking 2 semesters preparation for theory through math and experimental physics lectures (which of course I also attended ;-)).

In such a situation you have to use an "inductive" approach, often using the "historical approach", i.e., to follow more or less the development of the subject as it was in history, which of course has to be cleaned up by all the thorny failures in getting to the modern understanding.

After having learned the (theoretical) physics in the inductive approach, it's also good to have a deductive point of view, i.e., you start from some real "axioms" and develop the phenomenology from that. Then you can state the Newtonian postulates (I'd no call them axioms) simply by defining the spacetime model to be the Galilei-Newton spacetime, which is a bundle of Euclidean spaces along a directed 1D time continuum, leading to the notion of Newton's "absolute space" as being defined by the equivalence classes of inertial frames, which exist by assumption. In the taste of 20th century physics, based on 19th century geometric views culminated in Klein's Erlanger Programm, you can (and in my opinion) should analyze it using group theory and symmetry analysis.

So why is the lex quarta valid: Simply because it is hypothesized within the choice of a spacetime model! Now one can think, how to interpret Newton's famous saying: "Hypothesis non fingo" ;-))).

 

[Jiman]

Mistakes in Arnold? It's likely to have typos in any book, but real mistakes in Arnold? I doubt it.

Concerning the question about "Newton's axioms", one should know the history of Greiner's textbooks. They are meant to be used by students who start the general theory course already in the 1st semester, which was a novum at German universities at the time the books are written. I remember too well, how disappointed I was when I started to study physics at the university that theoretical physics was supposed to start in the 3rd semester only. But that were the good old times before the socalled "Bologna reform" on European universities, so you were free to do whatever you want, you only had to pass the "Vordiplom" exams after 2 years and the "Diplom" exams at the end. How you got the knowledge to do so nobody cared. So I went in the 3rd-semester theory lectures as a freshman. Greiner's books were among the best sources to read about all the many things I couldn't understand lacking 2 semesters preparation for theory through math and experimental physics lectures (which of course I also attended ;-)).

In such a situation you have to use an "inductive" approach, often using the "historical approach", i.e., to follow more or less the development of the subject as it was in history, which of course has to be cleaned up by all the thorny failures in getting to the modern understanding.

After having learned the (theoretical) physics in the inductive approach, it's also good to have a deductive point of view, i.e., you start from some real "axioms" and develop the phenomenology from that. Then you can state the Newtonian postulates (I'd no call them axioms) simply by defining the spacetime model to be the Galilei-Newton spacetime, which is a bundle of Euclidean spaces along a directed 1D time continuum, leading to the notion of Newton's "absolute space" as being defined by the equivalence classes of inertial frames, which exist by assumption. In the taste of 20th century physics, based on 19th century geometric views culminated in Klein's Erlanger Programm, you can (and in my opinion) should analyze it using group theory and symmetry analysis.

So why is the lex quarta valid: Simply because it is hypothesized within the choice of a spacetime model! Now one can think, how to interpret Newton's famous saying: "Hypothesis non fingo" ;-))).

Newton's parallelogram law is only mathematical method without physical significants,right?

 

[Jiman]

not three, more axioms, and they are independent
V Arnold Math methods of classical mechanics

Newton's parallelogram law is only mathematical method without physical significants,right?