It is known that a symplectic form is invariant along the trajectory of a Hamiltonian system. Based on this fundamental property, certain techniques have been developed. The aim of this paper is to extend such an approach to a wider class of dynamical systems, namely, generalized Hamiltonian systems. The authors consider a class of dynamical systems that possess a certain ``geometric structure''. Their results provide a theoretical basis for applying a symplectic algorithm to a considerably larger class of structure-preserving systems.