- #1

Theia

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

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?