So I have a very complicated Hamiltonian given by:
$$ H_R(\psi, P_\psi) = \frac{-B_0 R^3}{2 \pi} (2 \psi - sin(2 \psi) + \sqrt{B_0^6 R^6 V(\psi)^2 - P_\psi^2} $$
where $V(\psi) = \frac{1}{\pi} \sqrt{b_0^4 sin^4(\psi) + (\pi P/M - \psi + \frac{1}{2}sin(2\psi))^2}.$
$$V(\psi) = \frac{1}{\pi} \sqrt{b_0^4 sin^4(\psi) + (\pi P/M - \psi + \frac{1}{2}sin(2\psi))^2}.$$
The potential makes this Hamiltonian VERY hard (according to me) to evaluate. My goal is to get the equation of motion on form $\psi(R)=$ something, for arbitrary $P/M$. $B_0$ and $b_0$ are just some constants.
In principle I could start from $ \frac{\partial H_R(\psi, P_\psi)}{\partial P_\psi} = \partial_R \psi \equiv \dot{\psi}$.
$$ \frac{\partial H_R(\psi, P_\psi)}{\partial P_\psi} = \partial_R \psi \equiv \dot{\psi}.$$
Then I would get
$$ \dot{\psi} = \frac{-P_\psi}{\sqrt{B_0^2 R^6 V(\psi)^2 - P_\psi}}. $$
Then I could integrate $\psi(R)$ w.r.t $R$. But I don't know the limits of this integral. And even if I did, it would be a next to impossible task to integrate? Not even Mathematica can handle that integral.
So I will have to make some approximations I guess.
I could in principle solve it numerically but I want a solution for arbitrary $P/M$.
Any input on how I can ultimately get the equation of motion on form $\psi(R) = $ something?
I am grateful for any help.
$P_{\psi}$ is canonical momenta in $\psi$ which is the polar angle spanning a 3-sphere. P and M is the momentum and mass of a D-brane or graviton. The range of $\psi$ is $[0, 2 \pi]$
