1) Given the system dx dt dy dt xcos (xy) -ycos(xy) a) Show the system is Hamiltonian with H(x, y) = sin (xy) as it's fu

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1) Given the system dx dt dy dt xcos (xy) -ycos(xy) a) Show the system is Hamiltonian with H(x, y) = sin (xy) as it's function. b) Draw the phase plane (directional field using software) c) Find the solution curve that passes through the point (1,) and draw it on the phase plane from part b) (do not solve the equation for y or x) Your answer should include a screen shot of the phase plane and the solution curve.

2) Given the system a) Is the system Hamiltonian? b) If so, find the Hamiltonian function dx dt dy dt y X c) Draw the phase plane (directional field using software) d) Find the solution curve that passes through the point (1,2) and draw it on the phase plane from part c) (do not solve the equation for y or x) Your answer should include a screen shot of the phase plane and the solution curve.

3) Is the system Hamiltonian? If so, find a Hamiltonian function. If not explain why not. dx sin(x) cos (y) - cos(x) sin (y) dt dy dt

4) Is the system Hamiltonian? If so, find a Hamiltonian function. If not explain why not. dx sin(x) cos (y) 2x - cos(x) sin (y) dt dy dt

5) Given the system. dx dt dy dt y x² +a Where a is a parameter. a) Verify the system IS Hamiltonian b) Find the Hamiltonian function c) For what value/s of a does the system have one equilibrium point? d) For what value/s of a does the system have two equilibrium points? e) For what value/s of a does the system have no equilibrium points?