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

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?

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More on Newton's laws and superposition: physics.stackexchange.com/q/18080/2451 – Qmechanic Dec 18 '11 at 17:49
Isn't it a complete duplicate? – Ron Maimon Dec 31 '11 at 10:28

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. :)

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There is something good about this answer, its first sentence is quite true, and the rest is a useful commentary on it. But. It misses the point of the question in other ways. So I will attempt to supplement this answer with another answer. – joseph f. johnson Dec 31 '11 at 6:27
@Antillar: He is asking why forces combine like vectors. This is not answerable without some physics of forces, that they are the time derivative of a conserved quantity, and each component is conserved separately. This is adressed here: physics.stackexchange.com/questions/18080/… , so I don't want to repeat. – Ron Maimon Dec 31 '11 at 10:30
Yes, you are right, now I see the point you made in the other post, of which I was not aware. Vectors are only part of the story (why is force a vector in the first place?) and the explanation in terms of conserved quantities sounds perfectly fine. By the way, I'm afraid I didn't have a principle of superposition of quantum states in mind, I simply asked about the additivity of forces in newtonian physics. – quark1245 Jan 1 '12 at 13:51
"Why is force a vector?"- A very good question! I don't know enough group/field theory to give you an answer consistent with my previous answer. One does not have to explicitly speak of QM or CM when discussing superposition. It is a more general concept. – Antillar Maximus Jan 1 '12 at 15:51
Ron,thanks for pointing that out. Sometimes, I miss the obvious question. :) – Antillar Maximus Jan 1 '12 at 15:53

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.)

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-1: The question is asking "If I apply a 20N force and my friend applies a 30N force, is the total force exactly 50N? Why exact linearity?" The answer is yes, because it's the same as "I transfer 20 units of charge, and my friend 30, do I get exactly 50 units of charge?" Its true because momentum is an additive conserved quantity, a thing like charge, which you add up, and the force is defined as the change in this quantity, which then also exactly adds up. This linearity is exact, and this is the linearity the OP was asking about, not the approximate linearity in small-oscillation equations. – Ron Maimon Dec 31 '11 at 10:33
the principle of superposition is about states, it is not merely linearity in general. I paraphrased Dirac when I wrote « 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. » Adding forces or charges has nothing to do with the principle of superposition, so one answeres the OP by pointing out that the 'linearity' he asked about has nothing to do with the principle of superposition he asked about. – joseph f. johnson Dec 31 '11 at 16:46
you are criticizing his language then, not answering the question. The question is not about Dirac's superposition principle, but about additivity of forces. – Ron Maimon Dec 31 '11 at 16:57
If that were true, I would rewrite the answer to point out what « superposition » normally means since that would be instructive. But the OP accepted Antillar's answer which is along the lines of Dirac...so that contributed to my impression. – joseph f. johnson Dec 31 '11 at 17:10
Johnson, the linearity you are referring to is operator linearity (homomorphism + additional constraints). It does not make sense to talk about linearity of the spaces themselves. What do you mean by GR will necessitate modification of the superposition principle? Removed the -1 on Ron Miamon's comment. – Antillar Maximus Jan 1 '12 at 16:04