We investigate the multiplicity of $2{pi}$-periodic solutions of the nonlinear Hamiltonian system with perturbed polynomial and exponential potentials, $dot{z}= JG^{prime}(z)$, where $z:R{ ightarrow}R^{2n}$, $dot{z}={frac{dz}{dt}}$, $J=(array{0&-I\I&0})$, I is the identity matrix on $R^n,G:R^{2n}{ ightarrow}R$, G(0, 0) = 0 and $G^{prime}$ is the gradient of G. We look for the weak solutions $z=(p,q){in}E$ of the nonlinear Hamiltonian system.