The origin of conservation of momentum

[parsec]

I was having a discussion about the cause/s of conservation of momentum with a friend and was given a very unsatisfactory explanation (something to do with space being homogenous).

Is it an irreducable physical law like the conservation of energy, or is there a good (intuitive) physical explanation?

 

[CompuChip]

Try this approach:

Suppose we have two objects colliding elastically. They exert a force on each other: the first exerts a force F₁₂ on the second one, and the second one a force F₂₁ on the first one. By Newton's third law, F₁₂ = -F₂₁.
Add the two and use \vec F = \frac{\mathrm d\vec p}{\mathrm dt} to say something about the total change of momentum of both.

 

[D H]

Suppose we have two objects colliding elastically.

Elastic collisions assume conservation of linear momentum and conservation of energy hold, so using elastic collisions as the basis for deriving these conservation laws is a non-starter.

I was having a discussion about the cause/s of conservation of momentum with a friend and was given a very unsatisfactory explanation (something to do with space being homogenous).

Is it an irreducable physical law like the conservation of energy, or is there a good (intuitive) physical explanation?

Your friend was absolutely correct. That space is homogenous (space has translational symmetry) does indeed imply that momentum is a conserved quantity. Similar considerations lead to other conservation laws, including angular momentum (space has rotational symmetry) and energy (time translation). These are extremely deep and extremely powerful results.

All three conservation laws derive from Noether's theorem. Noether's thereom is a cornerstone of modern theoretical physics. Here are some links to articles on Noether's theorem: The Usenet Physics FAQ article http://math.ucr.edu/home/baez/noether.html" article, and
http://www.mathpages.com/home/kmath564/kmath564.htm" in Kevin Brown's math pages.

 

[CompuChip]

Ah yes, of course you are right. I was again trying to oversimplify the problem
Noether's law is a good idea

 

[Dale]

Of course, using translational symmetry to explain why there is conservation of momentum inevitably leads to the question why does the universe exhibit translational symmetry?

 

[Andy Resnick]

It's certainly possible to imagine a reality where an experiment performed *here* will lead to a different result if it is performed *there*, or that an experiment performed today will give a different result that if it were performed tomorrow. But that ultimately undermines causality and predictability; two of the cornerstones of science.

So, if the universe was not isotropic it's not clear there could be Science.

 

[tiny-tim]

I was having a discussion about the cause/s of conservation of momentum with a friend and was given a very unsatisfactory explanation (something to do with space being homogenous).

Is it an irreducable physical law like the conservation of energy, or is there a good (intuitive) physical explanation?

Hi parsec!

As D H says, Noether's theorem proves that any symmetry of a space results in a conservation law (strictly, a conserved current).

Translational symmetry of Einsteinian space-time implies conservation of the energy-momentum 4-vector.

Translational symmetry of Newtonian space-time implies conservation of the momentum 3-vector and either mass or energy (I forget which … and if it's only mass, then I don't know where Newtonian conservation of energy comes from, except as an approximation from Einstein. )

For an intuitive physical explanation of conservation of momentum:​

Suppose you have a set-up in which you find that energy and momentum happen to be conserved (perhaps for no reason … it could be of totally unconnected bodies that you just picked randomly).

Then you can work out that any other observer, with a different velocity from yours, using his measurements rather than yours, will also find that energy and momentum are conserved (this works for either Newtonian or Einsteinian geometry).

In Newtonian geometry, for example, if ∑m_iu_i = ∑m_iv_i, then ∑m_i(u - a)_i = ∑m_i(v - a)_i, for any relative velocity a.

And similarly for ∑m_i(u - a)_i².

This is geometry, not physics!

You can try other definitions of momentum, and you'll find none of them work in this way.

So geometry tells us that the same mass-and-velocity law can work for different observers, but only if it's conservation of momentum and/or energy.

I suppose you could have a universe in which there were no universal conservation laws (universal meaning that they work for observers with any velocity) … but if there are universal conservation laws, they must include momentum and energy! ​