Hamilton's Equations - Getting Period from quadratic solution

Question

If I have Hamiltonian's equations 

\(q(t)=H-\frac{t^2}{2}\)

\(p(t)=-t\)

How would I show that the motion is periodic and find its period?

Comments

They do not look like equations, but like solutions (with unknown letter $H$). If $H=E$, then it is a motion like a rolling down from the top of inversed parabole from $x(0)=E$ to $x(t\to\infty)\to -\infty$.