Explain for a Hamiltonian system why an equilibrium point is

Explain, for a Hamiltonian system, why an equilibrium point is a center for the nonlinear Hamiltonian system if it is a center for the linearized system at the equilibrium point. Given a nonlinear system dx/ dt =1-y^2 dy/dt = (1+y^2)sinx. Is the system (l)-(2) Hamiltonian? If it is, please find a Hamiltonian function, and describe the solution curves in the phase plane. Now, define a new system by dividing 1 + y^2. dx/dt = 1-y^2 / 1+y^2 dy/dt = sinx, Is the system (3)-(4) Hamiltonian? If it is, please find a Hamiltonian function, and describe the solution curves in the phase plane. Can you find solution curves of the system (l)-(2) in the phase plane? If yes, please describe them.

Solution