# Is Newton's first law a special case of a more general law?

I was reading my freshman physics textbook (fundamental's of physics by Jearl Walker), and the book says that Newton's first law only applies in a special frame of reference

Newton’s first law is not true in all reference frames, but we can always find reference frames in which it (as well as the rest of Newtonian mechanics) is true. Such special frames are referred to as inertial reference frames, or simply inertial frames.

What are frames of reference? What do they mean? I was not able to find a definition that I can understand at my current level. However, without knowing what a frame of reference is, I attempted to come up with another definition:

Assume that there's a set of all possible frames $$R$$, then we can rewrite the first law this way:

There exists a frame $$r \in R$$ such that, in the frame $$r$$ the following is always true: $$a = 0 \iff f = 0$$

What do you think of my definition? Can we find other frames with different laws? Can we prove they exist?

• Sep 28 '20 at 16:36
• Be a bit careful about trying to redefine physics in terms of math. Math is the language of physics, but math is NOT physics. Sep 28 '20 at 17:17
• @DavidWhite I'm having a hard time not thinking of physics as just math. Is there anything I can do about this? Maybe an example where thinking mathematically is not sufficient Sep 29 '20 at 15:51
• @user168651, for me, when I finally got enough experience to think in terms of concepts, I could "mix and match" those concepts until I was sure of the physics I was dealing with, then let the math flow from that. This means that I had to think above the math level, which is NOT an easy skill to acquire. Focusing on the concepts first, and the math second, is one way to see physics in a different light. Sep 29 '20 at 16:37

A reference frame is simply a system of co-ordinates measured relative to a specific point, which is the origin in that reference frame.

Often we use Cartesian co-ordinates in each reference frame (we don't have to, but this makes it simpler to define what we mean by a "straight line") and we rotate the co-ordinates in each reference frame so that the $$x,y,z$$ axes are aligned (again, we don't have to, but it makes life simpler). And we choose the origin in each reference frame so that all of the origins coincide at some specific time, which we call $$t=0$$.

We can then identify a particular point (or event) in spacetime by its co-ordinates and time relative to reference frame $$A$$ - say $$(x_A, y_A, z_A, t)$$. In another reference frame $$B$$ the same event will have different co-ordinates $$(x_B, y_B, z_B, t)$$. Note that because we are considering Newtonian mechanics here, the value of the time co-ordinate $$t$$ is the same in all reference frames - there is a universal time. If we were considering relativistic mechanics then $$t$$ would depend on the reference frame as well.

We can track the $$(x_A, y_A, z_A)$$ co-ordinates of some object $$O$$ in reference $$A$$ - in general these will depend on time $$t$$. If the $$(x_A, y_A, z_A)$$ co-ordinates of $$O$$ are constant (i.e. do not depend on $$t$$) then we say that $$O$$ is at rest relative to reference frame $$A$$. If the $$(x_A, y_A, z_A)$$ co-ordinates of $$O$$ depend linearly on time $$t$$ (so if $$x_A(t) = x_A(0) + vt$$ etc. ) then we say that $$O$$ is moving at a constant velocity relative to frame $$A$$.

By observing the co-ordinates of different events in reference frames $$A$$ and $$B$$, we can deduce a set of relations between the two sets of co-ordinates, and these relations hold for all events in spacetime. For example, if frame $$B$$ is moving relative to frame $$A$$ with constant velocity $$v$$ parallel to the $$x$$ axis then

$$x_A = x_B + vt \\ y_A = y_B \\ z_A = z_B$$

This is called a Galilean transformation. But if frame $$B$$ is accelerating relative to frame $$A$$ with constant acceleration $$a$$ parallel to the $$x$$ axis then

$$x_A = x_B + \frac 1 2 at^2 \\ y_A = y_B \\ z_A = z_B$$

and this is no longer a Galilean transformation.

If we have an object $$O$$ with no forces acting on it then we can define a reference frame $$F_O$$ in which this object is at rest (simply define the origin of the reference frame to be wherever that object is). Newtons' first law then says that any other object on which no forces act will either be at rest or will move with a constant velocity relative to reference frame $$F_O$$. And this will also be true in any other reference frame that is related to $$F_O$$ by a Galilean transformation.

However, Newton's first law will not be true in a reference frame that is related to $$F_O$$ by a non-Galilean transformation. In a reference frame that is accelerating relative to $$F_O$$ for example, then $$O$$ will appear to be accelerating even though there are no forces acting on it.

• Thank you! I really like the rigorous appraoch used here. Something that my physics textbook lacks. Are you aware of any introductory physics text book that follows a math-like approach? Sep 28 '20 at 15:27

Imagine 3 mutually perpendicular rigid rods, with markings at uniform intervals, extending to infinity. The rigid rods form a reference frame.

We can use the reference frame to describe the motion of any physical particle in space, by saying how the particle is located relative to the markings on the rigid rods at any particular time.

Now we can imagine two reference frames -- two sets of 3 mutually perpendicular, infinite rods. The two reference frames can be: (a) shifted relative to one other (the "origin" where the rods meet can be in different places) (b) rotated to relative to one other (the rods can be pointing in different directions) (c) moving relative to one other.

There will be some reference frames in which Newton's laws hold. What this means is that if you arrange your motion so that the rods of an "inertial reference frame" are not moving relative to you, then you will find that objects only move if a net external force is applied to them; that objects with mass $$m$$ will respond to an external force $$F$$ by moving with acceleration $$a=F/m$$; and that if object A exerts a force $$F$$ on object B, then object B exerts a force $$-F$$ on object A.