Hamiltonian structures of the first variation equations and symplectic connections

Necessary and sufficient conditions in terms of symplectic connections, ensuring that the first variation equation of a Hamiltonian system along a fixed invariant symplectic submanifold is also a Hamiltonian system with respect to some admissible symplectic structure are obtained. The class of admissible symplectic structures is distinguished by means of the natural condition of compatibility with the symplectic 2-form in the ambient space. Possible obstructions to the existence of a Hamiltonian structure on the first variation equation are investigated.