Register to reply 
Simple harmonic oscillator general solution 
Share this thread: 
#1
Sep2704, 09:52 AM

Sci Advisor
HW Helper
PF Gold
P: 4,771

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

HW Helper
Thanks
P: 10,767

ehild 


#3
Sep2704, 11:25 AM

Sci Advisor
HW Helper
PF Gold
P: 4,771

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

Sci Advisor
HW Helper
P: 3,144

Simple harmonic oscillator general solution



#5
Sep2704, 01:18 PM

HW Helper
Thanks
P: 10,767

ehild 


Register to reply 
Related Discussions  
Simple Harmonic Oscillator  Introductory Physics Homework  0  
1D simple harmonic oscillator  Advanced Physics Homework  3  
QM simple harmonic oscillator  Advanced Physics Homework  0  
Simple Harmonic Oscillator  Introductory Physics Homework  8  
Driven, damped harmonic oscillator  need help with particular solution  Advanced Physics Homework  1 