Clarke duality for Hamiltonian systems with nonstandard growth

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.