Henon-Heiles - Chaos

Systems that depart severely from integrability can only be studied numerically. An example of such a system is the 2-dimensional motion of a mass under the potential

$$V(x,y) = \frac{1}{2}( x^2 + y^2 ) + x^2 y - \frac{1}{3} y^3$$

This model was introduced by Henon and Heiles in 1964 to study stellar trajectories through a galaxy. This potential can be viewed as resulting from adding a (large) perturbation to the separable (integrable) harmonic potential. It is non-integrable.

The Henon-Heiles potential is zero at the origin, exhibits a triangular symmetry and shows ridges and valleys at large distances. All trajectories with total energies

$$E = \frac{m}{2}( v_x^2 + v_y^2 ) + V(x,y)$$

less than $1/6$ are bounded within the triangular region. The Maple worksheet Henon_Heiles_potential.mw illustrates this potential.

The equations of motion (Newton equation) lead to the ODEs

$$\frac{d x}{d t} = v_x$$

$$\frac{d y}{d t} = v_y$$

$$\frac{d v_x}{d t} = -\frac{1}{m} \frac{\partial V}{\partial x} = -\frac{1}{m} ( x + 2 x y )$$

$$\frac{d v_y}{d t} = -\frac{1}{m} \frac{\partial V}{\partial y} = -\frac{1}{m} ( y + x^2 - y^2 )$$

The trajectories lie on a 3-dimensional manifold in the 4-dimensional phase space. They are best studied via the $x=0$ SOS. To build this SOS you need to record the $y - p_y$ coordinates of each intersection of the trajectory with the $x=0$ plane. Further selecting the $p_x$ momentum sign, say when the trajectory goes into the $x=0$ plane, would normally be done. But,in this case,intersections with either signs are kept since the second choice corresponds to another existing trajectory. This stems from the invariance under $x \rightarrow -x$ and $p_x \rightarrow -p_x$.

The behavior of the system is determined by the only conserved quantity, the total energy $E$. Together with the SOS requirements, this points to the following sets of initial conditions: provide $E$, $x=0$, arbitrary $y$ and $p_y$, and calculate $p_x$ from the energy (selecting the positive value).

Plotting the trajectories in the laboratory frame, $y$ versus $x$, can produce beautiful pictures.

Comparing plots of $x$ or $y$ versus time in the regular and chaotic regimes is instructive.

Exercise

Solve the Henon-Heiles ODEs via Runge-Kutta 4th order

Implement the conservation of $E$ to provide a check on the accuracy of the solution.

Implement the initial conditions as described above via command line arguments.

Write a separate SOS filter code that projects the trajectory coordinates time series onto the $x=0$ plane. This code should read in the piped data from the ODEs solver code and output the time, $y$ and $p_y$ at the intersections. Use linear interpolation to refine the coordinates at crossings.

Explore the trajectories and SOS at various energies.