# How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it perturbatively, $$v=v_{(0)} +\lambda v_{(1)}+\lambda^2 v_{(2)}+\dots,$$ where $\lambda=\frac{bv_0}{mg}$ and $v_0$ is the initial velocity. Solving order by order we get $$v=v_0-gt+\lambda\left( -gt+\frac{g^2t}{v_0}-\frac{g^3t^3}{3v_0^2}\right)+O(\lambda^2).$$

Now we try to naively apply the method to a simple harmonic motion $$\ddot x+\omega_0^2x-\lambda\omega_0^2 x=0,$$ simply by expanding $$x=x_0+\lambda x_1+\lambda^2 x_2+\ldots.$$ With initial conditions $x(0)=A$ and $\dot x(0)=0$ we get the solution $$x(t)=A\cos(\omega_0t)+\frac{\lambda A\omega_0 t}{2}\sin(\omega_0 t)+O(\lambda^2),$$ which is non acceptable due to the linear (in time) growing secular term.

We solve it by improving the method. We allow the frequency to change, $$\omega=\omega_0+\lambda\omega_1+\lambda^2\omega_2+\ldots,$$ and use the Lindsted-Poincare trick. Then we get rid of the secular term and get the first correction in the frequency at once.

We knew that the solution with the secular term was physically non acceptable because we know the motion of the oscillator should be bounded. Moreover we can easily solve that oscillator since it is simply $$\ddot x+\omega_0^2(1-\lambda)x=0,$$ whose solution oscillates harmonically with frequency $\omega_0\sqrt{1-\lambda}$.

Question: Suppose I have a non linear system which I do not have enough intuition about their evolution (maybe I do not even know what physical system it describes). Then how do I know whether the regular perturbation theory (that naive one) will work? Is there any criteria telling whether I will have to extend the method (such as using "tricks" like the Lindsted-Poincare one)?

References: One of the nicest material at undergrad level I found is this lecture note. It discusses how the regular perturbation theory fails for an oscillatory motion. Yet it does not answer the questions I brought here. Another place I have found something in is Gregory's book on classical mechanics. However I found the discussion very short. Now I got a dover book about classical perturbation theory that looks quite nice but I am still at the beginning.

• There are lots of resources on classical perturbation theory on the internet. They are all truly horrible for my taste, though. Did you look at some of the horror? :-( May 12 '16 at 20:53
• @CuriousOne Yes I did! What I found was either too simplified (some people even do perturbation theory without explicitly expanding the solution in a perturbative serie. The results look like magic) or to complex (written by mathematicians with a background I don't have). May 12 '16 at 21:14
• I am glad I am not the only one who feels that way. Please let us know when you find a "classical perturbation theory for dummies" that we can all understand on some level. I need one, too. :-) May 12 '16 at 21:23
• @CuriousOne Please have a look at the references I posted. The lecture note is quite good! May 12 '16 at 21:33
• You are right, that's not a bad explanation, at least it makes things clearer along the lines of this example. A friend of mine who became a theorist once tried to give a lecture about the general method, half his listeners fell asleep, I guess, and a professor and his assistant stood up halfway trough the lecture and said "Please excuse us, but we know all of this... ". What he really meant is that even he couldn't make any sense of how my friend was trying to package what everybody who needed it intuitively knew about the procedure into a formal method, which was horribly complex. May 12 '16 at 21:40

One always needs to allow the frequency to change, otherwise one gets horrible secular terms. In case of resonances one needs additional tricks.

A good mathematical book is ''Perturbation methods in nonlinear systems'' by G.E.O. Giacaglia (Springer 2012). He discusses both the traditional Poincare-Linsted method and more advanced methods based on Lie transforms (canonical transformations). He also treats nonlinear resonances.

To check if a perturbative solution of a non oscillating system is good, one can use the optimized perturbation theory of

Stevenson, P. M. (1981). Optimized perturbation theory. Physical Review D, 23(12), 2916.

As long as varying parameters changes the results a lot, they are not yet good.

• Thanks for the reference, it looks quite interesting. Are you aware of any criteria (apart from convergence and asymptotic behaviors) to see if a perturbative solution of a non oscillating system is good? May 31 '16 at 23:45
• @Diracology: You can use Stevenson's optimized perturbation theory, which is very efficient as a check - as long as varying parameters changes he results a lot, they are not yet good. Jun 1 '16 at 4:44
• Stevenson, P. M. (1981). Optimized perturbation theory. Physical Review D, 23(12), 2916. Jun 1 '16 at 4:44