Hamiltonian structures of the first variation equations and symplectic connections

Necessary and sufficient conditions in terms of symplectic connections, ens uring that the first variation equation of a Hamiltonian system along a fix ed invariant symplectic submanifold is also a Hamiltonian system with respe ct to some admissible symplectic structure are obtained. The class of admis sible symplectic structures is distinguished by means of the natural condit ion of compatibility with the symplectic 2-form in the ambient space. Possi ble obstructions to the existence of a Hamiltonian structure on the first v ariation equation are investigated. Bibliography: 21 titles.