Looking for a serie solution for a nonlinear ODE system

[Theia]

Hi!

\(\displaystyle \begin{cases} \dot{q} = a \left( 1 - q^2 \right) \\ \dot{a} = - \alpha - a^2 q\end{cases} \qquad \alpha \in (0, 1 ) \)

I've looked into this ODE system about 7 months now, but I've not got anything promising how to write down the solution. I'm mostly interested in \(\displaystyle q\)-serie. (To those of you who are now thinking why I'm doing something like this: I'm too much interested in physics and projectile motion. This ODE system comes from quadratic air resistance when describing projectile motion. See e.g. this paper where to start.)

Methods tried thus far: Picard iteration, regular power serie method.

Picard iteration gives first powers very nicely, but it also shows how complicated the solution is. Also, I know that there are a lot of cool math hiding in the system. For example one can form a Lagrange equation (and Hamiltonian) to generate the system... In this case

\(\displaystyle \mathcal{L} = \frac{-1}{2} \frac{\dot{q}^2}{\left( 1 - q^2 \right)^3} + \frac{\alpha}{2} \left( \frac{q}{1 - q^2} + \artanh q \right)\)

Methods speculated: Perhaps it would help, if there was a way to write \(\displaystyle \artanh q\) in some kind of serie (definitely not like a power serie, but more in the spirit of Min-Max-approximation)...

Any thoughts or comments what to try?

 

[jvicens]

Hi there!

It seems like you have already tried some common methods for solving this ODE system. Have you considered trying numerical methods, such as Runge-Kutta or Euler's method? These methods may give you a more accurate solution compared to Picard iteration or power series methods.

Another approach you could try is to use a computer algebra system, such as Mathematica or Maple, to find a symbolic solution. These systems have built-in functions for solving differential equations and may be able to handle more complicated systems like the one you are working with.

As for writing \artanh q in a series form, you could try using a Taylor series expansion. However, keep in mind that this may not be a very useful approach since the series may not converge for all values of q.

Overall, I would recommend exploring different numerical and symbolic methods to see which one gives you the best solution for your specific problem. Good luck!