The solution to SHM Differential Equation is $y=A\cos wt + iB\sin wt$, upon applying this, where y would be angular displacement, how can an imaginary quantity pop up when applying this to the real world? for example, at time $t=\pi/4$, you get that the angular displacement is $0.5\sqrt2+i0.5\sqrt2$, how can the angular displacement be a complex number? How can you move $i$ radians around an axis?
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$\begingroup$ You might want to this is your solution makes sense given $y$ must be real... what about taking the real part on the right? $\endgroup$– ZeroTheHeroFeb 1, 2017 at 12:51
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$\begingroup$ how is that justified, to just omit the imaginary part? $\endgroup$– ThinkFeb 1, 2017 at 12:52
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$\begingroup$ because surely that would be a different solution of $(y=acoswt +bcoswt)$ without the i in the cos term. $\endgroup$– ThinkFeb 1, 2017 at 12:57
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$\begingroup$ @Think Try it without the i term. $\endgroup$– FarcherFeb 1, 2017 at 13:14
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2 Answers
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These are not the only solutions. You don't have to express solution in this way, you could just say that the solution is cos or sin without the imaginary unit $i$. BUT this solution you just wrote is a valid one. Physics do not have some kind of a monopoly to solutions of differential equations. Physicaly meaningful quantities in this solution you wrote are totally ok, you just ignore the imaginary unit and that is all.
So if you want to talk about the physics you just take the real part of the solution. If you plug in any number into your solution you get a complex number back, in general. But the value or modulus of this number is a real number as you can compute it using Pythagoras theorem.
So to sum up, in physics you are using mathematics to model reality. When you write something like this, using complex numbers, its just because it is easy to do it this way. You have to have in mind that you are taking just the real part because of course, physical quantities can not be imaginary. So if you have your oscillator with the amplitude A and some frequeny $\omega$, you can express it in many ways. The same motion.
In this case, what do you think the amplitude of the motion is? You can not add up real and imaginary number directly. You have to find the amplitude in some other way. And as I said you have to find modulus of your complex number. So complex solution is the most general one and physicist chooses how to describe particular motion. Maybe this would help: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/ComplexNumbersSHO.htm
So of course you can ask yourself what would be the motion of the point in the complex plane described by your general solution?
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We usually introduce the imaginary numbers because it is easier to calculate with them. It is easier to do powers of exponential than working with trigonometric transformations. But when your are done with your calculations you mast take the real part of your imaginary results and that would give the desired result. Transforming to the imaginary number is just computational convenience.
