Superposition principle and linearity

[tardon007]

hello folks!

this is my first post on the forums and I kick off with an interesting question...

I had been coming across the principle of superposition for quite some time and to admit frankly didn't ever understood it.

The most abstruse aspect is to comprehend how can to different inputs(take 'em as forces for the moment) set up out puts, totally ignorant of each others presence?

Can some one throw some light on this... Also , has the superposition principle been deduced emperically or has there been some physics behind the existence of the postulate...

 

[quantumdude]

Hi, and welcome to PF.

Superposition for forces is a direct consequence of the law of vector addition and of the linearity of Newton's law. That is, if you take your inputs to be forces (let's say there are 3 of them) and your outputs to be accelerations, what you have is the following:

F₁=ma₁
F₂=ma₂
F₃=ma₃

If you add these up to get the resultant force F, you get:

F₁+F₂+F₃

which of course is a vector sum. Now on the other side of the equation you get:

ma₁+ma₂+ma₃

Since Newton's second law is linear, we have:

F=ma,

where a is the resultant acceleration.

Superposition in other systems relies similarly on either scalar or vector addition, and on linearity of the system in question. If a system is nonlinear, superposition will not hold.

 

[Clausius2]

Hi tardón!

Superposition principle is a famous topic in many science areas: mechanics, electromagnetism, structural engineering, electrical engineering, electronical engineering and other parts that are governed by linear equations.

Surely, superposition principle had a mathematic fashioned way of being discovered. Former physicians knew about the linearity of the governing equations of some phenomenas. Although it is certain too that all this equations were built up over experimental researches. So that, former physicians must have had some empirical verifying of the superposition principle.

By the way, superposition principle is not employed at all in nonlinear sciences, like fluid mechanics, combustion theory, nonlinear physics and so on.