Conservation of Momentum from Newton's Laws

[Widdekind]

Can we confirm, that:

 

[math]\vec{F}_{12} = - \vec{F}_{21}[/math]

 

[math]\frac{d \vec{p}_1}{dt} = - \frac{d \vec{p}_2}{dt}[/math]

so that, integrated over time, w.h.t.:

 

[math]\delta \vec{p}_1 = - \delta \vec{p}_2[/math]

and so ensuring, that total momentum p₁ + p₂ stays the same?

[swansont]

Conservation of momentum is also a consequence of the symmetry of space translation, by Noether's theorem.

[Mr Skeptic]

Integrate it over time again, and center of mass is conserved.

[Widdekind]

Thanks !

[Oldman]

The sum of forces created by the constituents of a system within the system is zero. Similarly for torques. The universe is a system.

So, you have conservation of momntum, conservation of angular moentum, and Newton's third law.

[johnreed]

Recall that we have linear momentum and spin angular momentum from Newton’s first law. The angular velocity of a spinning disk, sphere, or solid object, is an artifact of the uniformly spinning disk, sphere, or solid object. The angular momentum of a spinning solid object also follows from the first law. We don’t have orbital angular momentum from that law.

 

We acquire orbital angular momentum from Newton’s mathematical derivation for orbital centripetal force, where he used a perfect circle and perfect motion to argue for orbital centripetal acceleration.

 

In other words Newton used the artifacts of spin angular momentum, ie. a perfect circle and perfect (uniform) motion, to argue the mathematical case for orbital centripetal force and orbital angular momentum. Here the only change in velocity is direction where the direction describes a perfect circle.

 

The spinning perfect circle angular velocity is an artifact of the uniformly spinning circle itself. So we have least action consistent, single object, spin angular velocity in all cases.

 

He then associated planet surface object mass [m] with orbital centripetal acceleration [v^2/r] by multiplying both sides of a least action consistent equation involving [v^2/r] by unity as mass/mass [m/m]. The product is [mv^2/r] and voila spin angular momentum was soon to became orbital angular momentum.

 

Newton then used the least action consistent angular velocity from Kepler’s empirical, time controlled law of areas describing 2 body planet orbital motion to mathematically carry his perfectly circular 2 body uniform motion, spin angular momentum analog, to the planet’s non-uniform 2 body orbital motion.

 

The generalization is based on least action consistent time-space parameters where the emergent conserved cumulative resistance of planet and moon surface atoms is either designated as the cause of the least action consistent celestial motion (Newton’s gravity), or as the consequence of the least action consistent 4D space-time curvature caused, continuum motion (Einstein and peers).

 

Regurgitation of original error.

johnreed