Problem 3: (i) Verify that the vector field F = (y, -:13) is Hamiltonian and find a Hamiltonian function for the field. (ii) Use the software of your choice to graph a few level sets of H (e.g. H = 1, 2, and 3).Problem 4: {i} Since the graphs of H = E {where E is a constant} are closed curyes we know the solutions are periodic. Lets determine the period of a solution with energy E and amplitude a. Use H[.r_.y] from the previous problem to write E in terms of o. Hint: based on the graphs of HLRy] = E the maximum displacement rr. happens when y = ii or at the point {n._.l]'}. {ii} The period of oscillation T can be found by evaluating a line integral of hit oyer the graph of H = E. That is T = fnir. But the first equation of our system tell us jgf = g, so by the chain rule air = \"f, and y '1" o: if. :r = / df = :1 r. .n . u .U Here we multiply by 4 because we're integrating over only the portion of the curve in the first quadrant {ti-at of a period}. Solve H{.f'.y]l = E for _y{.r} and use this to eliminate y from the above integrand. Then use the u-substitution u = g to simplify this integral. Write your answer as an expression depending on n times the integral jg: ' {iii} Is T increasing or decreasing as a increases