simple harmonic oscillator general solutionby quasar987 Tags: harmonic, oscillator, simple, solution 

#1
Sep2704, 09:52 AM

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In my mechanics textbook is given an exemple of how to find the general solution of the of the equation of motion for a force kx (the simple harmonic oscillator problem).
He begin his analysis and finds that e^(iwt) and e^(iwt) are both solutions. Hence C1*e^(iwt) and C2*e^(iwt) are also solutions and therefor C1*e^(iwt) + C2*e^(iwt) is also a solution and since it SEEMS to have 2 arbitrary constants in it, it could be the general solution. He then says that C1 and C2 MUST be complex in order for this to be the general solution. I'm guessing he's implying that if C1 and C2 are both real, then we can show that C1*e^(iwt) + C2*e^(iwt) turns out to have really just ONE arbitrary constant in it. Now let's try to do that. C1*e^(iwt) + C2*e^(iwt) = C1*[cos(wt) + isin(wt)] + C2*[cos(wt) + isin(wt)] = C1*[cos(wt) + isin(wt)] + C2*[cos(wt)  isin(wt)] (because cos(x) = cos(x) and sin(x) = sin(x)) = C1*cos(wt) + C1*isin(wt) + C2*cos(wt)  C2*isin(wt) = [C1 + C2]cos(wt) + [C1  C2]isin(wt) = C3*cos(wt) + C4*isin(wt) And this is two arbitrary constants. Does anyone sees the flaw... or has another idea? 



#2
Sep2704, 10:53 AM

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#3
Sep2704, 11:25 AM

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I think I understand.
OK, so strictly mathematically speaking, the statement that C1 and C2 are elements of R do NOT make for a general solution of the d.e. is WRONG. Any C1, C2 elements of the sedenions and beyond make up for a solution. But BECAUSE this equation represents a physical phenomenon, we are INTERESTED (the keyword missing in my textbook) only in those C1 and C2 that make x(t) REAL. And for that to be so, not only must C1 and C2 be complex, they must also be the complex conjugate of each other. Is that what you are saying? 



#4
Sep2704, 11:38 AM

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simple harmonic oscillator general solution 



#5
Sep2704, 01:18 PM

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