Mathematical model of Newton's first law

• I
vanhees71
Gold Member
2019 Award
That's not quite correct. The logical equivalent of the first expression is

${\rm If}\quad \frac{{d\vec v}}{{dt}} \ne 0\quad {\rm then}\quad \sum {\vec F \ne 0}$
Yes, and it's crucial that you mention that all your vectors refer to an inertial reference frame. The logic is that in Newtonian (as well as in special relativistic) physics there exist inertial reference frames and by definition (!) in this frames a free body moves uniformly (i.e., with constant velocity). This is the content of Newton's Lex I.

Then there's a definition of force in Newtonian physics which is the time derivative of momentum (which of course needs the introduction of mass beyond the kinematical quantities). This is Newton's Lex II.

Bestfrog
Yes, and it's crucial that you mention that all your vectors refer to an inertial reference frame.
Only if the third law is considered too. I already mentioned that the first and second law alone work quite well in non-inertial systems.

Then there's a definition of force in Newtonian physics which is the time derivative of momentum (which of course needs the introduction of mass beyond the kinematical quantities). This is Newton's Lex II.
That wouldn't exclude fictious forces. The full definition of force consists of all three laws of motion.

hclatomic
Is there a mathematic model for the first law of dynamics? If no, do you think that this law can be modellized with maths?
You quote an interesting point : Newton stated this shape of the force as a postulate. We are then driven to wonder if there could be a more mathematical explanation to it. Actually this is all the purpose of Lagrange's work (Mécanique analytique) which is marvelously explained in the Landau & Lifchitz "Mechanics". To be very straight forward they explain that the force is the derivative of the momentum (impulsion) with respect to time. As far as the momentum is $\vec P = m \vec v$, and the mass $m$ is a constant, we shall have $\vec F = m \vec a$. But please read the Landau & Lifchitz for more complete informations.

fedecolo
hclatomic
Please allow me to get forward, as I feel that your thought is in the right direction.

Newton is a great compilator, but he was an hawful guy. All people around him detested him. He finished his life away of every one, stuck by alchemy and horoscopes, far from sience. The famous $1/r^2$ gravitation law has been stated by Robert Hook, referencing the works of Huygens on the sling physics, in a letter to Newton. This last insulted him, saying that he was a freak and did not deserve any attention. But this made the reputation of Newton, the thief, in the "Principia", until us.

A particular experience is the debate with Emilie Du Chatelet about the kinetic energy, and this is a point close to your concerns. Emily shew experimentally that it is $mv^2$ and not $m v$ as Newton postulated (once again, postulate, postulate). Emily was right, Newton was wrong.

In the XIXth century, Lagrange entered in the same mood as yours. He felt that all these postulates should be explanable by the mathematics, in a way or an other. He wrote one of the most important piece of science of the humanity in "Mécanique Analytique", and bound everything and every one (Hyugens, Hook, Newton, Du Chatelet, ...) however by including the Maupertuis's postulate of least action. You heard of Hamilton, of course, but know trom now that this is only a rewrite of Lagrange's work in a particular mathematical way.

I really insist on this : read Lev Landau and Evgueny Lifchitz, Mechanics, Ed. Mir, Moscow. If you need to understand what the classical physics is, you will find no better way. All I told you here is mathematically stated in an elegant mathematical way by these authors.

Your questioning is at the door of the understanding, please get on.

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fedecolo
vanhees71
Gold Member
2019 Award
Only if the third law is considered too. I already mentioned that the first and second law alone work quite well in non-inertial systems.

That wouldn't exclude fictious forces. The full definition of force consists of all three laws of motion.
Of course, the 3rd law is implicit in the space-time model of Newtonian and special relativistic physics. That's why I didn't mention it. What you call "fictitious forces" (which I call "inertial forces") are no forces in the sense of interactions. You get the from bringing terms from the kinematics in non-inertial reference frames to the other side of the equation. That's all.

In general relativity the concept of the intertial frames become local, and you cannot distinguish gravity from inertial forces as far as local laws are concerned, but that's another topic.

Please allow me to get forward, as I feel that your thought is in the right direction.

Newton is a great compilator, but he was an hawful guy. All people around him detested him. He finished his life away of every one, stuck by alchemy and horoscopes, far from sience. The famous $1/r^2$ gravitation law has been stated by Robert Hook, referencing the works of Huygens on the sling physics, in a letter to Newton. This last insulted him, saying that he was a freak and did not deserve any attention. But this made the reputation of Newton, the thief, in the "Principia", until us.

A particular experience is the debate with Emilie Du Chatelet about the kinetic energy, and this is a point close to your concerns. Emily shew experimentally that it is $mv^2$ and not $m v$ as Newton postulated (once again, postulate, postulate). Emily was right, Newton was wrong.

In the XIXth century, Lagrange entered in the same mood as yours. He felt that all these postulates should be explanable by the mathematics, in a way or an other. He wrote one of the most important piece of science of the humanity in "Mécanique Analytique", and bound everything and every one (Hyugens, Hook, Newton, Du Chatelet, ...) however by including the Maupertuis's postulate of least action. You heard of Hamilton, of course, but know trom now that this is only a rewrite of Lagrange's work in a particular mathematical way.

I really insist on this : read Lev Landau and Evgueny Lifchitz, Mechanics, Ed. Mir, Moscow. If you need to understand what the classical physics is, you will find no better way. All I told you here is mathematically stated in an elegant mathematical way by these authors.

Your questioning is at the door of the understanding, please get on.
Thank you very much, I'm going to read this book immediately!

A Lazy Shisno
A particular experience is the debate with Emilie Du Chatelet about the kinetic energy, and this is a point close to your concerns. Emily shew experimentally that it is $mv^2$ and not $m v$ as Newton postulated (once again, postulate, postulate). Emily was right, Newton was wrong.
Eh, you've got a few misconceptions there. First, Newton didn't claim that kinetic energy itself was given by ##mv##; in fact, there was no distinction between kinetic energy and momentum during that time. It was Gottfried Leibniz who, as early as 1686, suggested that kinetic energy (which he called vis viva, or "living force") was proportional to the square of velocity. However, this view was widely contested by followers of Newton and DesCartes as it did not seem to be compatible with conservation of momentum, while ##mv## was (again there was no distinction between KE and momentum at the time).

It wasn't until 1719 and 1722 when Giovanni Poleni and Williem 's Gravesande independently confirmed Leibniz's quadratic relationship, empirically, by dropping balls onto clay. Williem 's Gravesande then told Emilie about his findings, who repeated the experiment, confirmed it, then published the results.

fedecolo