Can <PERSON>'s First Law be treated as a form of bias?
This is a quirky question, so bear with me. I'm fairly certain it is relevant. Also, I am using the term "bias" here very loosely. By that term, I basically mean any way in which a conclusion is affected by how data is collected, how it is modeled, or any other decision by an analyst (rather than by a feature of the actual underlying phenomenon). Now here's the question:
One way to formulate the basic problem of mechanics is to say that we want to model the position $S$ of an object or system of objects. Presumably $S\in\mathbb{R}^{3}$ for a single object or $S\in\mathbb{R}^{3n}$ for a system of $n$ objects. If $H\in\mathbb{R}^{m}$ represents the set of variables which are believed to affect the motion of our (collection of) object(s) then we want a model that looks something like
$S(t) = F(H;t)$
where the form of $F$ will not be specified. This could work, but $S$ depends on choice of origin. So no one should object if we instead model displacement
$S(t)-S(0) = F(H;t)$
which eliminates translational dependence on the coordinate frame. Now, maybe we run an A/B test where $H=H_{A}$ for the "A" test and $H=H_{B}$ for the "B" test. 474
Note that we will see no effect when
$F(H_{A};t) = F(H_{B};t)$
which amounts to
$S_{A}(t)-S_{A}(0) = S_{B}(t)-S_{B}(0)$
and dividing by $t$ gives
$\cfrac{S_{A}(t)-S_{A}(0)}{t} = \cfrac{S_{B}(t)-S_{B}(0)}{t}$
All well and good. But what if we only have observational data to work with? Unable to experiment, we will likely just have to compare different periods of time where different $H$ conditions are observed. Suppose $H_{A}$ was observed for the interval $(t_{A},t_{A}+\delta t)$ whereas $H_{B}$ was observed for the interval $(t_{B},t_{B}+\delta t)$. Well, now the equation is
$\cfrac{S(t_{A}+\delta t)-S(t_{A})}{\delta t} = \cfrac{S(t_{B}+\delta t)-S(t_{B})}{\delta t}$
which says that the two intervals should have the same average velocity. From this alone we already get the sense that no effect will be detected which does not change velocity. We can make this more explicit though. Set $t=t_{A}$ and $t_{B}=t+\delta\tau$. We get
$\cfrac{S(t+\delta t)-S(t)}{\delta t} = \cfrac{S(t+\delta t+\delta\tau )-S(t+\delta\tau)}{\delta t}$
Doing a little more simple math
$\cfrac{S(t+\delta t+ \delta\tau )-S(t+\delta\tau)} {(\delta t)(\delta\tau )} -\cfrac{S(t+\delta t)-S(t)} {(\delta t)(\delta\tau )} = 0$
in the limit $\delta t,\delta\tau\rightarrow 0$ this is
$\text{acceleration} = 0$
so, to recap, no effect will be detected if it does not accelerate the object or at least one of the objects in the collection, and this followed straight-forwardly from the choice to model displacement. Now, it is not clear to me whether physicists have a single clear definition of force, but it seems like "something that affects motion" is a pretty general description thereof. And, from this simple argument, it would seem that no force should be detected when modeling constant-rate displacement. This looks suspiciously like <PERSON>'s First Law, but it results simply from the choice to model displacement rather than from any underlying quality of nature. 272