- #1
giulioo
- 7
- 0
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 ;)
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 ;)