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

Question

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

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

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 $[0, 2 \pi]$

Answer 1

from the Hamiltonian :

$$H=H(\psi,P_\psi)$$

you get two first order differential equations :

$$\dot{P}_\psi=-\frac{\partial H}{\partial \psi}=f_1(P_\psi\,,\psi)\tag 1$$ $$\dot{\psi}=\frac{\partial H}{\partial P_\psi}=f_2(P_\psi\,,\psi)\tag 2$$

if you linearized Eq. (1) and (2), you obtain analytical solution.

$$\begin{bmatrix} \Delta{\dot P}_\psi \\ \Delta{\dot{\psi}} \\ \end{bmatrix}=\left[ \begin {array}{cc} {\frac {\partial }{\partial P_{{\psi}}}}f_{ {1}} \left( P_{{\psi}},\psi \right) &{\frac {\partial }{\partial \psi} }f_{{1}} \left( P_{{\psi}},\psi \right) \\ {\frac { \partial }{\partial P_{{\psi}}}}f_{{2}} \left( P_{{\psi}},\psi \right) &{\frac {\partial }{\partial \psi}}f_{{2}} \left( P_{{\psi}}, \psi \right) \end {array} \right]_{[P_\psi=P_{\psi_0},\psi=\psi_0]} \,\begin{bmatrix} \Delta{ P}_\psi \\ \Delta{{\psi}} \\ \end{bmatrix}$$