Construct a Poincaré section for the Hénon–Heiles potential. It is suggested that you make...

Construct a Poincaré section for the Hénon–Heiles potential. It is suggested that you make the plot in the plane so that you can compare your results with Figs. 11.7 and 11.8. Choose an energy, E, and initial conditions, x = 0 and , and initial conditions on y and to satisfy the energy condition and find the boundary curve. Relax the condition on and choose conditions on , y, and that satisfy the energy condition for x = 0. Integrate the equations of motion to find the crossings.

(a) Choose , y0 = 0.01, , and x0 = 0. Use the energy equation to determine .Integrate the equations to find the values of t where x(t) ≈ 0 saving the values . Find the first 27 crossings and compare with Fig. 11.7.

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(b) Repeat this process for and plot the chaotic behavior.