1Ph 106bHomework Assignment No. 4Due: Thursday, 7 February 20081. Regular motion in the H´enon-Heiles Potential.Consider a particle moving in two dimensions, governed by the HamiltonianH =12(p2x+ p2y)+V (x, y) ,where V (x, y) is the H´enon-Heiles potentialV (x, y)=12(x2+ y2)+x2y −13y3.a) Numerically integrate the equations of motion on the energy surface H = E =1/12, withinitial datax(0) = −0.1 ,y(0) = −0.2 ,py(0) = −0.05 .Plot x(t) and y(t) for time t in the interval (0, 400).Hint: Use the software of your choice. If you use Mathematica, the NDSolve commandcreates interpolating functions x[t] and y[t], which can be plotted using Plot[Evaluate[x[t]]]and Plot[Evaluate[y[t]]]. It is recommended that you use Mathematica version 5.2 rather thanversion 6.b) Plot the Poincar´e section of your solution from (a) on the two-dimensional slice throughthe energy surface at x = 0, showing points where the trajectory passes through theslice for time t in the interval (0, 1000). Choose the coordinates (y, py) on the slice.Hint: If you use Mathematica, the ParametricPlot3D command can plot the points in anarrow slice near the x = 0 surface.2. Irregular motion in the H´enon-Heiles Potential.The same as problem (1), but now with energy E =1/8.23. Elliptic and hyperbolic fixed points of a two-dimensional map.Consider the two-dimensional mapM :x → x0= x + y,y → y0= y + f (x + y) ,where x and y are real numbers. Here, f is a differentiable function, its only zero is at theorigin, f (0)=0, and its derivative at the origin is f0(0) = K.a) Express the 2 × 2 first derivative matrix of M in terms of the derivative of f . Is the mapM area preserving?b) Find the unique fixed point of M . For what values of K is the fixed point elliptic? Forwhat values of K is it hyperb olic?4. Poincar´e-Cartan theorem.The Poincar´e-Cartan theorem (see page 212 of Ott) asserts that for two closed curves Γ1andΓ2that enclose the same “tub e” of trajectories in (2N +1)-dimensional extended phase space,IΓ1ω =IΓ2ω,where ω is the one-formω = pidqi− Hdt .To prove the theorem, we use Stokes’ theorem:IΓ1ω −IΓ2ω =ZΣdω ,where Σ is a two-surface along the tub e with boundary ∂Σ=Γ1− Γ2, anddω = dpi∧ dqi− dH ∧ dt .It remains to show that the integral over Σ vanishes. For this purpose, we parametrize Σ withvariables (s, t), where s labels a trajectory in the tube, and t is the time along the trajectory,and then “pull back” the two-form dω to the (s, t) space. Writingdpi=∂pi∂sds +∂pi∂tdtdqi=∂qi∂sds +∂qi∂tdtdH =∂H∂pi∂pi∂sds +∂H∂qi∂qi∂sds +∂H∂pi∂pi∂tdt +∂H∂qi∂qi∂tdtand using Hamilton’s equations, complete the proof of the Poincar´e-Cartan theorem. (Recallthat the wedge pro duct is antisymmetric: ds ∧ ds =0=dt ∧ dt and ds ∧ dt = −dt ∧