Inertial and Non-Inertial Frames and the Galilean Transformation
Turns out Newton's first law isn't obvious at all β it's secretly guaranteeing that inertial frames exist, and that's the whole setup for the Galilean transformation.
Back in General Mechanics I, I kept hearing that this chapter is the most important one β and also not at all a breeze.
Someone once called it “the first obstacle course of the general mechanics textbook,” chapter 5… (sigh)
It’s about rotation, which middle school and high school didn’t exactly beat into us until we were sick of it. So yeah. It’s hard. It is.
Ughhh I don’t know I don’t know, it’s so haaard….
OK, let’s start here:
“What is a non-inertial frame of reference?”
Before we can answer that, we need to pin down what an inertial frame of reference actually is.
Spoiler up front: an inertial frame is a frame in which Newton’s second law holds.
That’s… it. Spoiler over.
The inertial-frame story actually starts with Newton’s first law. So let’s walk through it.
First Law: the Law of Inertia.
In school they teach it like this: “Objects have a property called inertia.” Plus examples β tripping over a stone, falling backward in a bus that suddenly takes off.
Seems way too obvious, right?
But surely there’s more going on here. So let’s push.
The first law is, yes, about inertia β that part’s correct. Stated a little more generously: if no external force acts on an object, its motion doesn’t change. A thing at rest stays at rest; a thing in motion keeps moving. Stretch it one more step β no external force means zero acceleration.
But… this feels so tangled up with the second law. (Honestly, I think I was the same way) β in school, didn’t you first understand the second law, and then backfill the first law from there???
So why on earth did Newton call this the first law and acceleration the second?
Because what the first law is really saying is this:
“There exists a frame in which, when there is no external force, the acceleration is zero.”
(Otherwise F=ma just… doesn’t work.) (Put another way: unless you’re sitting in that kind of frame, F=ma won’t work unless you tack on an inertial-force term.)
(cf. The frame where acceleration is not zero even though there’s no external force? That’s the bus frame. Person in the bus suddenly gets slammed backward and falls over… meanwhile the bus is just standing there like, “bro β why is this guy falling? I’m not even moving.”
(((You’ve only really gotten what a frame is once you properly understand the sentence “the bus is still” in that setup.)))
And another thing β think about it for a sec. When we set the origin of the “inertial frame,” where do we set it? Wherever I’m standing, which is… the Earth, right? But the Earth is not an inertial frame. It’s rotating, it’s revolving, it’s accelerating every single moment.
The reason we get away with treating Earth as inertial is that the effects of its rotation and revolution are just so, so, so tiny. heh.
(Since Earth came up β why does the first law need to guarantee “such a frame exists”? Because in a place that’s doing rotational (accelerating) motion like Earth, Newton’s laws straight up don’t hold. They just don’t match reality, so Newton patches it over with “inertial force.”
And then a problem pops up: “am I rotating right now, or am I not rotating?”
This is normally where you start talking about absolute space β the story that begins with the bucket thought experiment.
But I’ll just plant a flag here and step away.
…Yeah… this whole thing… Newton’s absolute space, absolute time… caused endless arguments even after Newton, starting with the constancy of the speed of light popping out all clean in Maxwell’s electromagnetism, which made people start squinting at Newton,,,, and then Einstein just demolished the whole setup. Basically.
But as everyone knows, Newtonian mechanics is still the tool that approximately and conveniently describes the physics we actually run into every day. So properly understanding it is definitely not a waste….)
OK back to it!
So there exists such a frame, we define that frame first, and then we talk about the relationship between force and acceleration.
F=ma. (2nd law.)
And as a property of force, every force has an actionβreaction pair. (3rd law.)
But isn’t the 3rd law also kinda vague? Like, it sounds insanely obvious?
The way the 3rd law should actually be read is:
“If there’s an action and there’s no reaction, then that’s not a force.”
I’ll just touch on that and keep moving. Newton’s laws really seem to be the hardest things in the world to state properly…..
Fun fact: the Principia was originally classified not as physics but as philosophy. Makes sense (sigh)(sigh)
Anyway, what I was actually trying to get at here β the inertial frame. Did you catch it?????
Oho… then NOW β what is a non-inertial frame?!?!
It’s just a frame in which Newton’s second law doesn’t hold.
Concretely (and as I hinted earlier), “an accelerating frame” is the textbook example.
In an accelerating frame, the second law naturally doesn’t hold. So β to protect Newton’s law β we introduce an inertial force term.
And in this post we’ll look at a simple inertial-force term! Not the one from rotation β just the one from plain old straight-line motion.
Which means we need the Galilean transformation.
First: “transformation” basically means “change your point of view.”
FYI, the Galilean transformation isn’t the only transformation in town. In relativity, the Lorentz transformation shows up.
Both of them are about rotating the inertial frame (= your point of view) β but they differ in what they preserve.
Galilean preserves $x^2 + y^2 + z^2$. Since Newton treats time as absolute, it’s just… left out of the picture entirely.
Lorentz showed up to fix the contradictions that blow up in relativity β so it adds time, and the preserved quantity becomes $x^2 + y^2 + z^2 - t^2$. That’s the thing called Lorentz invariance.
(I covered this in an earlier post β Special Relativity #3, spacetime and Lorentz invariance. I said at the end of that one I’d come back with a non-Euclidean geometry angle… and yeah, I will, just not yet…)
Everything inside the dotted lines up there is purely context. I don’t think you need to dwell on it in classical mechanics. I just wanted to flag that this thing exists, and hopefully it makes the rest click a little easier…. heh heh.
In the Oxyz coordinate system, point P’s position is $\vec{r}$. In the moving O’x’y’z’ coordinate system, point P’s position is $\vec{r}'$.
Then the link between the two coordinate systems, using $\vec{R}_0$, is:
$$\vec{r} \;=\; \vec{R}_0 \;+\; \vec{r}'$$Now differentiate that equation, twice in sequence:
$$\vec{v} \;=\; \vec{V}_0 \;+\; \vec{v}' \\ \vec{a} \;=\; \vec{A}_0 \;+\; \vec{a}'$$V and A are the velocity and acceleration of the moving coordinate system. v and a are point P’s velocity and acceleration as seen from Oxyz. v’ and a’ are point P’s velocity and acceleration as seen from O’x’y’z’.
Important case:
$$\vec{A}_0 \;=\; 0$$What if this? That means the moving frame is in uniform motion or at rest. Then:
$$\vec{a} \;=\; \vec{a}'$$Take some object at P with mass m β ma = ma’, so F = F’.
And what does that mean? It means the force F seen from the stationary frame (a physics student in a stationary room) and the force F’ seen from a frame moving at constant velocity (a physics student in a room cruising along at constant velocity) are exactly the same!!
But if the acceleration of the moving frame is
$$\vec{A}_0 \;\neq\; 0$$then
$$m\vec{a} \;=\; m\vec{A}_0 \;+\; m\vec{a}' \\ \vec{F} \;=\; m\vec{A}_0 \;+\; \vec{F}'$$F and F’ are different!!!
A discrepancy shows up between point P’s F as seen by Oxyz and point P’s F’ as seen by O’x’y’z’. The discrepancy is $m\vec{A}_0$, and that term is called inertial force.
This thing isn’t a force that arose from some actual physical cause β it’s purely a discrepancy that pops up because you changed perspective. So calling it a “force” is already a stretch. Which is why “fictitious force” might be the more honest name. haha haha haha.
One more thing, because this part is incredibly funny….. A is outside the bus, B is inside the bus. The bus goes vroom! and accelerates, and B falls over. Why does B fall? Inertial force, we say.
But from B’s point of view, A also suddenly goes vroom! in the other direction! So why doesn’t A get an inertial force???
Newton says: because A is the absolute standard, and B is the one moving relative to it.
But then β somewhere in the universe there has to be something that is the super-absolute standard, right? Newton says yeah, some absolute reference frame exists, it’s just… unmeasurable. Cue the bucket thought experiment from earlier…
If I keep pulling on this thread I’m going to end up saying every single thing there is to say about Newtonian mechanics in one post… I’ll study up more and maybe do a Newtonian-mechanics special feature or something down the line.
Anyway β Newtonian mechanics is definitely not worthless, so this post was basically a warm-up for the next one. Revisiting high school Newtonian mechanics, tightening the whole picture…
And in the next post: let’s actually wrap our heads around inertial force in the context of rotation! haha haha haha. See you there!
Originally written in Korean on my Naver blog (2015-06). Translated to English for gdpark.blog.