# Why does inertia happen?

In this video R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true?

• Well, we can say that it happens because the laws of physics are space-translation invariant. This is the part where you ask why the laws of physics should have that property. Then we just shrug and say "Because they do.". – dmckee --- ex-moderator kitten Jun 28 '13 at 18:40
• @dmckee Using that theorem, isn't energy the conserved quantity resulting from time-invariance? So given that mass is the same thing as energy, inertia is saying that the quantity of space-translation invariant stuff is proportional to the time-translation invariant stuff. I'm sure there's something really deep behind that... but I don't know what. – Alan Rominger Jun 28 '13 at 21:35
• – Qmechanic Jun 28 '13 at 21:40
• @dmckee How does space-translation invariance imply inertia? – isomorphismes Jun 30 '13 at 18:17
• @Qmechanic I know what inertia is, so I don't think the questions are similar. – isomorphismes Jun 30 '13 at 18:18

A way to see that is Noether theorem, which states that, if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

More precisely, consider a Lagrangian of a particle, moving in one dimension $x$, subject to a constant force $F$:

$$L = \frac{1}{2} m \dot x^2 + Fx$$

Imagine now the transformation $x \rightarrow x + a$, where $a$ is a constant

You see that $L$ transform as $L \rightarrow L + Fa$

So, if $F = 0$, the Lagrangian is invariant by the transformation $x \rightarrow x + a$, so this transformation is a continuous symmetry for the Lagrangian, and then there is a conserved quantity in time, which is simply the momentum $p = m \dot x$. You have $\dot p = 0$

If $F \neq 0$, the transformation $x \rightarrow x + a$ is no more a symmetry for the Lagrangian, and the momentum $p$ is no more conserved, and you have $\dot p = F$

• So if I know/accept/understand the conservation of momentum, do I learn anything new by learning the concept of "inertia"? Or do they refer to the same thing? – isarandi Jan 24 '15 at 11:55

Similar questions are: "why does electric charge happen?" and "why does gravity happen?" etc.

The "art" of physics is in the identification of the fundamental "stuff", stuff for which the question "why" is actually misguided.

You see, if there are fundamental "things" then, by the definition of "fundamental", these are the givens that we accept without question. For, if these fundamental things can be explained, they aren't fundamental.

Now, a reasonable question is this: is inertia fundamental?.

I don't know.

• +1, I upvoted, although I don't agree with the last asserion that "Is it fundamental?" would be a "reasonable question", whatever that means. If I had to somehow order question w.r.t. their reasonableness, I'd always prefer the aspect which depend on the human condition. E.g. "Is the concept of inertia useful for people who get paid to do physics? Can you make simple, efficient working theories in using it, one which other people can understand within reasonable time?" (and maybe arguing via algorithmic information theory) – Nikolaj-K Jun 28 '13 at 23:16
• Thanks for answering. I guess I was looking for a more mathematical answer. What would inertia being "fundamental" mean? – isomorphismes Jun 30 '13 at 22:14