For the damped harmonic oscillator equation$$\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x=0$$we get that the general solution is$$x(t)=Ae^{-\gamma t}e^{i\omega_d t}+Be^{-\gamma t}e^{-i\omega_d t}$$where $\gamma = \frac{c}{2m}$ and $ \omega_d=\sqrt{\omega^2-\gamma ^2}$. Using Euler's equation, we can expand this as follows:$$Ae^{-\gamma t}(\cos(\omega _dt)+i\sin(\omega_d t))+Be^{-\gamma t}(\cos(\omega _dt)-i\sin(\omega_d t))$$$$\Rightarrow e^{-\gamma t}(A+B)\cos(\omega_d t) +e^{-\gamma t}(Ai-Bi)\sin(\omega_d t)$$But now we are dealing with a physical problem so we only examine the real part which is $e^{-\gamma t}(A+B)\cos(\omega_d t)$. But this does not have any phase difference. Yet textbooks always make the claim that the real part of the solution is$$e^{-\gamma t}(C)\cos(\omega_d t+\phi)$$where $\phi$ is some arbitrary initial phase. But where does that initial phase come from if the real part of the solution does not have a phase change in it? I understand that $A$ and $B$ themselves need not be real however I do not understand how this fact could ever lead to a non zero initial phase in the real part of the solution.
This issue has bothered me for quite some time now so any help would be immensely appreciated!
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$\begingroup$Your split into real and imaginary parts is flawed for generic complex coefficients A and B. In general, you may parameterize them as$$ A=a e^{i\theta+i\phi}, \qquad B=b e^{i\theta-i\phi}, $$for real a,b,θ,φ. It is evident that θ is superfluous, as its corresponding phase factors out of the linear equation, and thus x.
So only φ is meaningful, and it joins up with the dynamical phase $\omega_d t$, providing a new origin for it. Then, with real a,b, the amended real-imaginary split you attempted would be sound, and the conventional result would follow for real C=a+b.
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