17 June, 2026
Surface of section
I am currently working on the analysis of the phase space. For this I am wanting to understand the outer-most trajectory in the surface section with the condition \(x=0\) and \(p_x\geq 0\).
Our first constraint comes from the energy of the system.
\[\HH(x,y,p_x,p_y)=\Ekin(p_x,p_y)+\Epot(x,y)=E\]
To do this, the equation
\[\Epot(x,y)= \VHH \]
is considered for \(x=0\) and \(p_x=0\).
\[ \begin{align} \Epot(y)\equiv\Epot(0,y) &= \frac{m \omega^2}{2}y^2 - \frac{\alpha}{3}y^3 \\ \Ekin(p_y)\equiv\Ekin(0,p_y) &= \frac{p_y^2}{2m} \end{align} \]
so we consider the energies that pass through the surface section \(x=0\) with maximal kinetic energy in the \(y\) component, i.e. \(p_x=0\) and \[p_y=\pm\sqrt{2(\HH-\Epot(0,y))}.\]
These trajectories move through \(x=0\).