Newtons three laws of motion from a single rule.

[Riad]

Dear all
I have come with a simple idea to generate the three laws of motion of Newton and would love to hear comments on its correctness.
Assume equal point masses: the rule says ' the displacement of one point mass must be balanced by an equal and opposite displacement of another'. Since all masses are identical, the displacement of ten points one unit distance can be balanced by the displacement of one point ten times the distance.. or the 'sum of mass x displacement is zero everywhere'. That is the centre of mass of an isolated system never moves. Now differentiate wrt to time and you get conservation of momentum and diff again and get the conservation of force- ie action and reaction at any point and in any direction is equal.
What is interesting about this is:
1- the centre of the whole universe never moves-accepted fact. 2-inertia have simple explanation. 3- ideas like the entanglement of particles across large distances can be accommodated- since we do not require particles to be at the same position.

 

[Dale]

This works, but only in the reference frame where the center of mass is at rest.

 

[Cleonis]

Dear all
I have come with a simple idea to generate the three laws of motion of Newton and would love to hear comments on its correctness.

I agree that you present a way of showing there's redundancy in the way Newton's three laws are formulated.

I think it's worthwile to also state the following assumption explicitly:
Axiom: The universe is uniform in space and in time.

This uniformity isn't necessarily the case, and according to the general theory of relativity there is in fact curvature of spacetime. So to formulate Newtonian mechanics that axiom is necessary.

Arguably, Newton's first law serves the same purpose as the unifomity axiom. The first law demands that objects in free motion are in uniform motion, that happens if and only if space and time are uniform.

Seen in this way you do need more than one law to generate Newtonian mechanics. These laws can be formulated in a number of ways, and it's interesting to compare different formulations.

 

[Riad]

I agree that you present a way of showing there's redundancy in the way Newton's three laws are formulated.

I think it's worthwile to also state the following assumption explicitly:
Axiom: The universe is uniform in space and in time.

This uniformity isn't necessarily the case, and according to the general theory of relativity there is in fact curvature of spacetime. So to formulate Newtonian mechanics that axiom is necessary.

Arguably, Newton's first law serves the same purpose as the unifomity axiom. The first law demands that objects in free motion are in uniform motion, that happens if and only if space and time are uniform.

Seen in this way you do need more than one law to generate Newtonian mechanics. These laws can be formulated in a number of ways, and it's interesting to compare different formulations.

Many thanks for taking the time to comment. I personally think there is a misunderstanding of the curved space concept. I understand that Einstein is not calling for a bent space, but bent space-time which is different. Space alone is always flat.

 

[Riad]

I agree that you present a way of showing there's redundancy in the way Newton's three laws are formulated.

I think it's worthwile to also state the following assumption explicitly:
Axiom: The universe is uniform in space and in time...

I also agree with you that Newton's first law in the form 'if something is moving in a straight line then it will continue to do so' might be violated in a curved space-time setting. However if you put the additional requirement that 'unless disturbed' then this will take account of the curved space-time.. since this means the presence of other masses that will disturb this body motion.
If we follow the balanced displacement rule above, then the body will continue in a straight line unless we move another body such that the sum(massxdisplacement)=0. Note the force that causes the motion is not mentioned.. ie how much a point moves is decided by other laws.

 

[Riad]

I agree that you present a way of showing there's redundancy in the way Newton's three laws are formulated.

I think it's worthwile to also state the following assumption explicitly:
Axiom: The universe is uniform in space and in time...

There is an additional assumption though.. the superposition of vectors- normally accepted as the one important property of our 3D space.

 

[Cleonis]

I personally think there is a misunderstanding of the curved space concept. I understand that Einstein is not calling for a bent space, but bent space-time which is different. Space alone is always flat.

About spacetime curvature:

Among the early explorations of Einstein was a theory in which space is as linear as in Newtonian dynamics, and only time is affected. This turned out to be a dead end.

GR predicts a deflection of light that passes by the Sun, the closer the passage the stronger the deflection. (But still only a fraction of a degree.) The deflection predicted by GR is twice as large as the prediction of the earlier exploratory theory. The difference comes down to the space-curvature component of the spacetime curvature.

So there is no room for an interpretation that space is always linear, and that the spacetime curvature is introduced by way of time effects only.

[later edit]
Light takes in a unique position in that the way a gravitational field deflects it is effectively in equal parts due to space curvature and time dilation effects.

At non-relativistic speeds the space curvature effects are far smaller than time dilation effects. Einstein's early exploratory theory, that had gravitational time dilation but no space curvature, did pretty good for the orbits of the planets. (It may even have reproduced the Newtonian orbits exactly, but I can't remember that detail.)
[/later edit]

 

[Riad]

About spacetime curvature:

Among the early explorations of Einstein was a theory in which space is as linear as in Newtonian dynamics, and only time is affected. This turned out to be a dead end. ..

I very much appreciate your reply.
The way I understand it is if v=0, ie no motion, then relativity becomes the same as classical mechanics.. this is like taking a tangent to the curved space-time to get a flat space. Or you can take a section of space time at constant (t) - a still picture if you like. Does this not support my conclusion that space on its own is always flat.

 

[Cleonis]

The way I understand it is if v=0, ie no motion, then relativity becomes the same as classical mechanics.. this is like taking a tangent to the curved space-time to get a flat space.

Well, while you can always define a tangent space that is geometrically flat, that has no bearing on the space itself.

Simplest example: a circle. If you divide that circle in 6 sections then you find that each section curves 60 degrees. As you divide in more and more sections the curvature per section becomes less and less, but not zero.

For a circle the definition of the amount of curvature is as follows: the ratio of the section curvature and the section length. In the limit of taking infinitisimally short sections both section length and section curvature tend to zero, but their ratio is a non-zero number; the actual curvature of the circle.

 

[Riad]

Well, while you can always define a tangent space that is geometrically flat, that has no bearing on the space itself...

I have a feeling that there is a difference here. There is a reduction in dimensionality- the t dimension is removed- a still picture.. may be it is like looking at the circle sideways and seeing a line rather than a circle. Of course there is a difference here, the remaining dimensions are three not two. Somehow the word flat is not the best word to describe a 3D space.
Would you be kind enough to direct me to literature regarding Einstein's first attempts about this matter.

 

[Cleonis]

Would you be kind enough to direct me to literature regarding Einstein's first attempts about this matter.

I'm not aware of a publication that discusses differences between Einstein's 1911 explorations and the 1915 GR.

On his personal website Eric Baird offers a translation of http://www.relativitybook.com/resources/Einstein_gravity.html" . That establishes from source that the 1911 exploration led to a deflection prediction of 0.83 seconds of arc.

Both the 1911 exploration and 1915 GR are implementations of the Principle of Equivalence, that is why the gravitational time dilation in the two frameworks comes out the same. Yet 1915 GR predicts twice the deflection. This illustrates that the space curvature that is part of 1915 GR is an essential feature.About using Physicsforums:
It's good that in your replies you cut away paragraphs, leaving only what needs to be quoted. However, you need to preserve the closing tag. Markup tags are in between squared brackets "[]", and a closing tag has a forward slash in it "/". In order to be formatted correctly a quoted paragraph has a starting tag and a closing tag.

 

[Riad]

Many thanks for the valuable information and sorry about the wrong cutting in your article. I am a newish user, so forgive me.

 

[Cleonis]

Both the 1911 exploration and 1915 GR are implementations of the Principle of Equivalence, that is why the gravitational time dilation in the two frameworks comes out the same. Yet 1915 GR predicts twice the deflection. This illustrates that the space curvature that is part of 1915 GR is an essential feature.

Einstein's book, Relativity: the special and general theory, is available at wikisource.org .

In appendix III http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Appendix_III" are discussed

Einstein remarks:
"[...] according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification (" curvature ") of space caused by the sun."

I need to mention that "produced by the Newtonian field of attraction" is meant metaphorically here. Strictly speaking classical theory predicts zero deflection of light, as classical theory is limited to gravitational attraction between chunks of matter. A classical evaluation consists of calculating the deflection of a material object that is traveling at the speed of light.

The underlying idea is that the gravitational time dilation aspect of spacetime curvature can be regarded as the relativistic counterpart of the Newtonian field of attraction.

The slower the velocity of a celestial body, the smaller the contribution of space curvature in its orbit. That is why for planets further away from the Sun than Mercury the perihelion precession is even smaller than that of Mercury.