Where does the principle of superposition come from in newtonian mechanics?

Question

Part of the definition of the concept of force is that if particle $1$ exerts a force $F_1$ on particle $3$ and particle $2$ exerts a force $F_2$ on particle $3$, the total force on particle $3$ is $F_1+F_2$.

But, is the principle of superposition deducible from Newton's laws or is it an additional assumption? If so, is it always valid? Is this fact linked to the non-existence of three-body forces (or do such forces exist?) or to some kind of linearity in the laws of mechanics or to some kind of fundamental symmetry?

Answer 1

The principle of superposition arises from the fact that we use Vector spaces to describe Physics. The abstract organization goes something as follows:

1. Start with a set S, of mathematical objects.
2. Organize them as a Group (call it a "Vector" Group). A group is your original set S and an operation (say multiplication).
3. Now bring in a Real (number) Field. A Field is a Set and two operations (the set and either of the operations form groups).
4. Make the Vector group and the Real Field talk to each other by defining a few rules. The resultant is a Vector Space over a Field.

The simplest case is to choose operations that are common to both the Field and the Vector group.

When we write $\vec{v}=a\hat{x}+b\hat{y}$, $(a,b)$ are elements of the Real Field and the "unit vectors" are abstract elements in the Vector Group. The $+$ sign arises because we define how elements in the Field ought to talk to elements in the Vector Group, and also how this new animal talks to other such animals.

The point and power behind an abstract formulation is that the moment you cast a physical object as a Vector, all rules/theorems/results that are applicable to vector spaces are automatically valid (Edit: you will find out very quickly if you started off with the wrong idea). You essentially reduce the problem from one of organization to one of interpretation.

This may seem rather abstract and difficult to understand, but I can assure you that it is not! If it seems difficult, it is only because I have done a poor job of explaining things. :)

Answer 2

In classical Physics, it is the theory of waves and the wave equation that has a principle of superposition, not Newton's second law as you are wondering about.

Why Newton's second law is irrelevant: In fact, it isn't linear in gravity: the Force law is $1\over r^2$ which leads to non-linear behaviour. The principle of superposition in QM says something totally different than adding forces as you are asking, it says that if a system can be in state $v$ or it could be in state $w$, then it is also possible for it to be in the state $2v+3w$. This links up to Antillar's answer: the principle of superposition says that the possible states form a vector space. But the positions of the planets in Newtonian graviation, or in Einstein's theory either, do not form a vector space.

But this principle does come from classical Physics, it comes from wave motion: the equation of a vibrating string, $$ -{\partial^2 f\over \partial x^2} = {\partial^2 f\over \partial t^2}$$ for the height $f$ of a string above the $x$-axis above the point $x$ at the time $t$. This equation is a linear equation: the set of solutions is a vector space: if $f$ is one solution and $g$ is another, then $2f+3g$ is also another possible solution.

But this wave situation is much more special than Newton's Law, although Newton's Law is used in proving the wave equation. So the principle of superposition cannot be deduced from Newton's law at all. It is now thought to be a universal principle which is exactly true, even though now QM kind of abandons Newton's Laws, the principle of superposition is now even more important than it ever was in classical Physics. (In classical Physics it was not universal.)

(I suppose one could worry about whether General Relativity's being non-linear means that eventually we will have to modify the principle of superposition.)