In my study of Hamiltonian dynamics I have come across a Hamiltonian dynamic system with a solution curve I know to be closed via computer and via intuition but I require a rigorous way to prove this, the problem simply is:
We have the curve in $\mathbb{R}^2 $ described via the equation $\sin(x)\sin(\pi y)=1/2 $ (of course for single period of the sines as they are periodic so this equation can describe infinitely many curves). We are to show this curve is closed. The Hamiltonian is $ H(x,y)=\sin(x)\sin(\pi y) $
My intuition was regarding periodicity of the sines. My purpose is basically to show it is not a limit cycle but a closed orbit in phase space, so can someone please demonstrate rigorously why this curve is closed? I have the intuition but not the right rigorous ideas. I thank all helpers.