I was reading a research paper titled "Dynamical analysis of bounded and unbounded orbits in a generalized Hénon-Heiles system." The link to the paper is here and on arXiv. In this paper, the authors study how adding an additional term to the Hénon-Heiles potential influences the orbits of objects.
In the last part of the second section of the paper, the authors find the energy thresholds for which the motion becomes unbounded ($E_{min}$ and $E_{max}$). This Wikipedia article: https://en.wikipedia.org/wiki/Escape_velocity says that an object escapes, or the object's orbit becomes unbounded, when the sum of the kinetic and potential energy is zero. I have understood this has the total energy has to be zero. However in Table 1 in the paper, the authors are getting nonzero energies. Could someone please explain what is going on?
Best Answer
In this answer I wont tell you what you need to calculate exactly, but I try to show you how you can come up with the right ideas yourself.
What you need to understand is the "potential-landscape" of the discussed model. So if you consider a point particle moving in a $2$D potential $V(x,y)$, you can imagine the particle as moving on a table which is shaped like the graph of the potential in a uniform constant gravitational field. So if this potential has a local minimum the particle might be trapped in such a minimum. Unless the particle has enough additional kinetic energy to escape the trap. Of course the amount of kinetic energy needed depends on the depth of the local minimum and the initial conditions.
So given that the initial position is "inside" a neighborhood of the local minimum at $(x,y) = (0,0)$ with total energy $E$. The largest amount of potential energy the particle could have is given by $V(x,y) = E$ (Hence the kinetic energy vanishes). So if the energy is low enough the contour given by $V(x,y) = E$ is closed and the particle is trapped in the enclosed region, so the orbit is guaranteed to be bounded. But if the potential has a shape which allows for channels through which the particle can escape for a high enough energy, the contour is no longer closed around a finite region and the trajectory is not necessarily bounded. Note however that bounded orbits are still possible but not guaranteed.
Let us now consider the potential in your paper for $\delta = 0$ $$ V(x,y) = \frac 1 2 (x^2 + y^2) + x^2 y- \frac 1 3 y^3, $$ which is plotted below.
So below you see two pictures of the contour $V(x,y) = E$, which is just the cross section of the potential above with a surface parallel to the $x-y$-plane at height $E$, where on the top $E < 0.1\bar6$ and on the bottom $E > 0.1\bar 6$.
So you see the contour around the local minimum is closed on the top, whereas there are channels through which the particle can escape on the bottom.
In the paper you mention the authors distinguish between total energies for which one can escape through at least one channel $E > E_\text{min}$ and energies for which one can escape through all three channels $E > E_\text{max}$. For $\delta = 0$ it holds $E_\text{min} = E_\text{max}$ but for $\delta \neq 0$ the energies are not the same. You can "explore" this by looking at the contour plots just as I have shown you above. (For example using Mathematica, or something similar)