Perhaps a trivial question, but it is something which I couldn't ever grasp ever since beginning physics. Why exactly should Newton's second law be linear in application of all the external forces?
For example, suppose I have a spring oscillating with a block (in absence of the gravity of Earth) then I add in gravity, then simply to find the new governing equation for acceleration I include gravity into the equation for the second law… but why? Why is there no cross effect between the two 'forces' acting on the block?
To be clear, I do not mean that force is linear in the sense of it being a linear function of time or a linear function of position. I know very well that forces could manifest whatever complicated function you can imagine, and in certain cases, as a series expansion.
My question asks about the linearity in terms of the application of different causes. As described by the example, let's add some more complication, let's say the Moon suddenly became 'massive enough' that its effect could be felt on our spring system i.e: the effect is no longer non-negligable for wherever I am on Earth conducting the experiment. Then the new governing equation of acceleration, again, would just be given by including the force of the Moon in my old force sum.
$$ a = \frac{\sum F_{ext} }{m} \to a'= \sum \frac{ \sum F_{ext} }{m} + \frac{F_{moon} }{m}$$
And, sure, a direct answer may be "because force is modelled by a vector and vectors add in so-and-so fashion", but, in physics we are trying to model what we see in the real world rather than impose our mathematical truths on it. Hence, again I ask, are there any deep reasons why in the real world, the forces (causes) of motion add up independently (without having a mixing effect)?
Best Answer
Note: I have edited my original answer to take into account some comments and to add more generality, in particular stressing the superposition principle as a more general principle than the pair-wise additivity, the latter being a particular case of the former.
I wouldn't call linearity the property you are referring to. The proper name superposition principle. In the simplest case, the superposition principle coincides with the pair-wise additivity of the interactions, i.e., we assume that if the force on body $i$ due to body $j$ alone is ${\bf F}_{ij}$, and that due to body $k$ alone is ${\bf F}_{ik}$, the total force in $i$, due to the simultaneous presence of $j$ and $k$, is $$ {\bf F}_{i} = {\bf F}_{ij}+{\bf F}_{ik}. $$ Notice that in the separate case and in the combined case, each pair-wise force (${\bf F}_{ij}$ and ${\bf F}_{ik}$) is a function only of quantities of the corresponding pair. More in general, for an n-body system, $$ {\bf F}_{i} = \sum_{j=1;j\neq i}^n{\bf F}_{ij}. $$ Superposition (or pair-wise additivity) is definitely a mathematical property different from the bi-linearity of the vector sum. The former has to do with the functional dependence of the contributions to the force on the body parameters, the latter with the operations defined on these functions.
Superposition, or the more specific pair-wise additivity of the forces, is often used in Newtonian mechanics and it was taken for granted by Netwon, but it is not a necessary condition. Indeed it is quite easy to provide examples of more complicated forms of force law. Even more important, although rarely stressed in the textbooks, the most accurate models of the effective forces among atoms or molecules in condensed matter are certainly not pair-wise additive (see the comment at the end).
Probably the most simple example of a force that is not pair-wise additive is the force between two neutral but polarizable particles, say $1$ and $2$. If only these two particles are present the mutual forces are zero: ${\bf F}_{12}=0$ and ${\bf F}_{21}=0$. However, if we introduce a third, charged particle say number $3$, both the original particles get an induced electric dipole and, in addition to the dipole-charge interactions with the charged body, ${\bf F}_{21}\neq 0$ and ${\bf F}_{12} \neq 0$, due to the dipole-dipole interaction.
A couple of final comments are in order: