Definition. A vector field F(r, y) is called Hamiltonian if there exits a function H (called the Hamiltonian function) such that F(x, y) = (H,(x, y), - H,(x, y)) . Definition. A vector field F(x, y) is called conservative if there exists a function f (called a potential function) such that fA = ( ( 1 0 ) ff ( no ) f ) = (NI)A Compare the two definitions. Both define the field F in terms of the partial derivatives of a scalar function; Hamiltonian functions are to Hamiltonian vector fields as potential functions are to conservative vector fields. In fact we could write the Hamiltonian vector field using the same "del" operator F(x, y) = J( VH) where J is the rotation matrix J = This J matrix is the same rotation operation that we used to define the normal vector in 2d, J((I, y) ) = (y, -I) and N ( t ) = Ji" (t ).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 o. Use H[.r._.y}i from the preyious problem to write E in terms of o. Hint: based on the graphs of H{.r__y] = E the maximum displacement o happens when y = [i or at the point {o..f]|}. {ii} The period of oscillation T can be found by evaluating a line integral of in? over the graph of H = E. That is T = fart. But the first equation of our system tell us 3% = y, so by the chain rule df = , and 5r Ll" od- T=/dr=-1 r. .n .o .7} Here we multiply by 4 because we're integrating over only the portion of the curve in the first quadrant {H4 of a period}. Solye Him. y} = E for yim} and use this to eliminate y from the above integrand. Then use the usubstitution u = j to simplify this integral. Write your answer as an expression depending on :1 times the integral f' f" r U e'Iu': I {iii} Is T increasing or decreasing as .1 increases