Momentum as a conserved measure of motion

Useful, but not really fundamental?

So far momentum looks like just one measure of motion amongst many (velocity, energy in the kinetic store) – useful for some facets, less useful for others. With the approach we've adopted so far, there seems nothing very fundamental about the quantity – it just so happens that mass  ×  velocity is a useful measure of motion – one that you can change by the accumulated action of a force acting on the object over time. However, it's a bit more fundamental than that.

There is a very deep mathematical theorem, that connects the conservation of momentum – that is, that momentum of an isolated body must be conserved – to space being homogeneous. So if you move from one place to another, and all other things stay the same, the world will behave in the same way: the rules of the universe are invariant under translation. This is an example of a symmetry argument, which is increasingly commonly in modern physics, yet rare in elementary physics teaching.

It's one of three such fundamental principles, which you can caricature as:

• It doesn't matter when you are, the rules of the universe should be the same.
• It doesn't matter where you are, the rules of the universe should be the same.
• It doesn't matter which way you're facing, the rules of the universe should be the same.

In the early 1900s, Noether showed (much more rigorously than we can dream of doing here) that these three constraints lead to the principles that:

• The conservation of energy is a necessary consequence of invariance under translation in time.
• The conservation of momentum is a necessary consequence of invariance under translation in space.
• The conservation of angular momentum is a necessary consequence of invariance under rotation in space.

So it looks as if the momentum of an object ought to always be conserved, unless you exert a force on that object. This applies to absolutely anything that can be seen as an object – anything where we don't have to worry about the internal structure or internal interactions, but can reduce the thing to a single point. (Remember the kinds of simplifications made in the SPT: Forces topic?) Such things might include cars, elephants, galaxies, trains, a volume of air containing millions of particles and maybe even a rowing eight.

Exerting a force on the object, so modelling its interactions with other objects, will change the momentum: it's only for objects isolated from their environment, and with no resultant force acting on them, that the momentum is conserved.