When particles physically interact, they transfer linear momentum and angular momentum between one another via force .
When particle P1 exerts force on particle P2, P2 exerts an equal and vectorially opposite force on P1 . That way, any momentum leaving P1 is attached to P2, and total momentum is conserved.
"equal and opposite force" is a necessary and sufficient condition for conservation of linear momentum.
But linear momentum is uncaring about the direction of the force, so long as it is equal and opposite.
If P1 and P2 were on opposite sides of a chosen origin point, at x,y (1,0) and (-1, 0) , and P1 put a +Y ( clockwise ) force on P2, the vectorially opposite -Y force on P1 on the other side of the origin point would also be clockwise around the origin.
That would perfectly satisfy "equal and opposite", and therefore satisfy conservation of linear momentum, but would violate conservation of angular momentum. Particles in our universe cannot do this.
Equal and opposite forces along a single straight line do not create torque or modify total angular momentum when viewed from any arbitrary origin point .
"Equal and Opposite force, along the path of separation" ( attraction or repulsion only ) is a necessary and sufficient condition for conservation of Angular Momentum to all origin points .
( It is possible to choose origin points to ignore various torques . But true conservation of Angular Momentum works from any and all origins )
So, from this point of view, it appears as if the more stringent "Equal and Opposite force, along the path of separation" of Angular Momentum automatically enforces the less stringent
"Equal and Opposite force" of linear momentum .
So, is conservation of linear momentum a simplified consequence of conservation of Angular momentum ?
Here is a messy supporting argument :
If a finite number of particles were interacting in a finite sized cube ( perhaps 1 meter across ), and you chose an origin point at z = 1 million meters, and you measured the
angular momentum around your chosen origin, you could get a pretty accurate estimate of the total linear (x,y) momentum of those particles by dividing the measured angular momentum by the
vector to the center of the box. You could make your estimate arbitrarily accurate by increasing your distance.
Conservation of angular momentum from this far origin point would imply that linear momentum was arbitrarily close to being conserved.
I have never seen thes arguments stated anywhere in print or elsewhere.
As an undergraduate circa 1981, I briefly showed this argument to my favorite physics professor, who off-the-cuff said he thought that this was a known fact. I was confused as to why it was not covered in the Halliday and Resnick.
So, is this a correct or incorrect concept ?
Can anyone find a hole in my not-terribly-complicated math ?
Does it appear in current textbooks ?
Best Answer
Conservation of linear and angular momentum arise from fundamental symmetries of spacetime via Noether's theorem. Conservation of linear momentum comes from translational symmetry and conservation of angular momentum come from rotational symmetry. The two symmetries are not necessarily related so the two conservation laws are not necessarily related.
Your argument is based upon the notion that if we consider the entire system then both types of momentum are conserved, which is equivalent to saying that no external forces or torques are acting. But this is simply the statement that spacetime, i.e. the background on which the bodies move, obeys both symmetries. This is certainly true of flat spacetime but not necessarily true of curved spacetimes i.e. not necessarily true of the universe as a whole, though I concede that it does appear to be true for at least the bit of the universe we can see.
But in practice we are often considering systems in which we have a background field, for example motion in a $1/r$ potential ($1/r^2$ force) such as the gravitational field of a spherical mass. You would argue that this is not an inertial frame since we are taking the spherical mass to be fixed, and this is certainly true. However it is still the case that the symmetries of the system determine the conservation laws i.e. in this case the rotational symmetry means angular momentum is conserved while the lack of translational symmetry means linear momentum is not conserved. To argue that the two conservation laws are equivalent would be to ignore a property of the system that is very important for performing calculations.