Simple Harmonic Motion--Quadratic Perturbation
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
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5-7-2018
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Simple Harmonic Motion--Quadratic Perturbation
Given a simple harmonic oscillator with a quadratic perturbation, write the perturbation term in the form 
,
 |
(1)
|
find the first-order solution using a perturbation method. Write
 |
(2)
|
and plug back into (1) and group powers to obtain
 |
(3)
|
To solve this equation, keep terms only to order 
and note that, because this equation must hold for all powers of 
, we can separate it into the two simultaneous differential equations
Setting our clock so that 
, the solution to (4) is then
 |
(6)
|
Plugging this solution back into (5) then gives
 |
(7)
|
The equation can be solved to give
![x_1=(alphaA^2)/(6omega_0^2)[3-cos(2omega_0t)]+C_1cos(omega_0t)+C_2sin(omega_0t),](http://mathworld.wolfram.com/images/equations/SimpleHarmonicMotionQuadraticPerturbation/NumberedEquation6.gif) |
(8)
|
Combining 
and 
then gives
where the sinusoidal and cosinusoidal terms of order 
(from the 
) have been ignored in comparison with the larger terms from 
.
As can be seen in the top figure above, this solution approximates 
only for 
. As the lower figure shows, the differences from the unperturbed oscillator grow stronger over time for even relatively small values of 
.
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