  We consider the existence of periodic solutions to Hamiltonian Systems with
growth conditions involving G-function. We introduce the notion of symplectic
G-function and provide relation for the growth of Hamiltonian in terms of
certain constant $C_G$ associated to symplectic G-function $G$. We discuss an
optimality of this constant for some special cases. We also provide an
applications to the $\Phi$-laplacian type systems.
