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

			(4)
			(5)

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

(8)

Combining and then gives

			(9)
			(10)

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 .