Forced oscillations and resonance (bis)

[giulioo]

I'm studying from landau lifšits "mechanics". I had some troubles in section small oscillations-->forced oscillations, especially from eq 22.4 to eq 22.5

i searched the web and came across this:

https://www.physicsforums.com/threads/forced-oscillations-and-ressonance.488538/#post-3236442

this thread does not answer the questions i had. Now I know the answer but the thread is closed and I cannot reply. Therefore i write it here (hoping this is the correct place and that it will be helpful to someone). I'm answering question n 2) since the physical meaning of beta is (as any initial phase) just a traslation in time (question n 1) ).

first of all the general solution to equation 22.2 and 22.3 is 22.4:

$$ x=a \cos(\omega t+ \alpha) + \frac{f}{m (\omega^2-\gamma^2)} \cos (\gamma t +\beta)$$

L&L rewrites this in the form

$$ x= a' cos(\omega t + \alpha) + \frac{f}{m(\omega^2-\gamma^2)} [\cos (\gamma t +\beta)-\cos(\omega t+\beta)]$$
where
$$ a'=\frac{f}{m (\omega^2-\gamma^2)} \frac{\cos \beta}{\cos \alpha} + a $$

(note that a' is a function of $\gamma$)
L&L does not write $a'$ but writes again $a$ (which is confusing). Now from this general solution he takes only the second addend as a particolar solution. then he takes the limit for $\gamma \rightarrow \omega$:
$$ \lim_{\gamma \rightarrow \omega} \frac{f}{m(\omega^2-\gamma^2)} [\cos (\gamma t +\beta)-\cos(\omega t+\beta)] = \frac{f}{2m\omega}t \sin(\omega t + \beta) $$

This is a special solution of $\ddot{x}+\omega^2 x = f/m \cos(\omega t+\beta)$ (in the limit $\gamma \rightarrow \omega$). The general solution is the sum of a special solution plus the general solution of the homogeneus equation associated, which is just formula 22.5 :

$$x(t)= a \cos(\omega t+ \alpha) + \frac{f}{2m\omega}t \sin(\omega t + \beta) $$

remark: in this formula $a$ is the same as eq 22.4 (is a constant and is not $a'$).

Hope someone will get benefit from this as i would have had yesterday ;)

 

[giulioo]

sorry my bad, at line "This is a special solution..." i meant $\ddot x+ \omega ^2 x =f/m \cos(\gamma t+\beta)$ instead of $\ddot x+ \omega ^2 x =f/m \cos(\omega t+\beta)$.