Given a simple harmonic oscillator with a quadratic perturbation, write the perturbation term in the form ,
(1)
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find the first-order solution using a perturbation method. Write
(2)
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and plug back into (1) and group powers to obtain
(3)
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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
(4)
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(5)
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Setting our clock so that , the solution to (4) is then
(6)
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Plugging this solution back into (5) then gives
(7)
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The equation can be solved to give
(8)
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Combining and then gives
(9)
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(10)
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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 .