Why are linear problems "Solvable"

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Why are linear problems "Solvable"

Postby polymer » Fri Oct 15, 2010 7:22 am UTC

This is something that has been on the back of my mind for a fair bit. Awhile ago I was reading the Feynman lectures, and Feynman pointed out that a large variety of problems were linear, and that mathematic's flexibility and power is really rather limited except when linear systems are being discussed. I found the thought interesting, and decided to pay attention in the future to when linear things popped up. To my surprise I started seeing these situations everywhere... F=ma was the first that came to mind, d=rt is an important relationship to. Linear combinations of the various "dimensions" of sine and cosine can describe periodic functions and sound with ease. Springs( or linear approximations) are everywhere in physics... And just recently, when studying probability the book tells me that one of the Axioms is basically a linear relationship (The probability of the union of mutually exclusive events is just the sum of the probability of the individual events.) It isn't clear to me why f(a + b) = f(a) + f(b) makes things so solvable, but I keep seeing it all over the place. I did take linear algebra, but I haven't vested too much energy into it personally though, so it may just be a basic disconnect between subjects. Regardless the question still bothers me and I'm curious, what makes linear relationships so special?

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Re: Why are linear problems "Solvable"

Postby wozub » Fri Oct 15, 2010 4:58 pm UTC

Well it makes things easier to analyze.

For one, when things are linear you get the principle of superposition. That lets you break complex systems into little parts and analyze those individually.

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Re: Why are linear problems "Solvable"

Postby samk » Sun Oct 17, 2010 1:30 am UTC

Nonlinear systems can have many areas of parameter space that seem to be heading towards a solution but aren't. Search local vs global optimum.

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Re: Why are linear problems "Solvable"

Postby RogerMurdock » Tue Oct 19, 2010 4:14 am UTC

Well perhaps the reason we study those things you listed so much is because they are linear?

If all springs exhibited crazy nonlinear motion, they probably would be studied only in specialized scenarios when necessary. Imagine fluid flow was suddenly very linear and easy to study. You'd be in here like "wtf fluid flow is everywhere and it's all we do jeez".

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Re: Why are linear problems "Solvable"

Postby polymer » Tue Oct 19, 2010 8:09 am UTC

Thanks to everybody who responded so far.

wozub: The principle of superposition is a really nifty tool. I especially like it over orthogonal basis's :). Seeing a periodic function decomposed into a sine and cosine waves is probably my favorite application of the principle so far.

samk: Thanks for the terms, I'll be sure to google them!

RogerMurdock: excuse my fascination that a common trick happens to be a really common and useful tool :p. The fact that F=ma is a linear relationship in particular is fascinating/curious to me since that equation is a fundamental relationship in physics. This being the case with E&M and Quantum as well is all the more fascinating. I respect and understand that lifting a pendulum slightly higher then 30 degrees, or slapping a second pendulum at the end of a first makes the problem far more complicated then your standard conservation of momentum problem. But that doesn't change the fact that these fundamental relationships, and many others, don't have to get much more complicated before they get unwieldy.

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