How do I get the equations of motion starting from this Hamiltonian?

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

$$V(psi) = frac{1}{pi} sqrt{b_0^4 sin^4(psi) + (pi P/M – psi + frac{1}{2}sin(2psi))^2}.$$

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

$$ frac{partial H_R(psi, P_psi)}{partial P_psi} = partial_R psi equiv dot{psi}.$$

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 $[0, 2 pi]$