  In this paper, the general planar piecewise smooth Hamiltonian system with
period annulus around the center at the origin is considered. We obtain the
expressions for the first order and the second order Melnikov functions of it's
general second order perturbation, which can be used to find the number of
limit cycles bifurcated from periodic orbits. Further, we have shown that the
number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) &
\mbox{if}~y>\varepsilon f(x)\\ (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x)
\end{cases}$ equals to the number of positive zeros of $f$ when at
$\varepsilon=0$ the system has a period annulus around the origin.

Dynamical Systems