Hamiltonian formalism for fiberwise linear dynamical systems

We study the Hamiltonization problem for fiberwise linear dynamical systems on a vector bundle in a wide class of symplectic structures. Two types of results are presented: (i) a Hamiltonization criterion is formulated as the solvability of some differential equations on the base including a matrix equation of Lax's type; (ii) a geometric interpretation of these equations is given in terms of symplectic connections. We consider some examples where there are nontrivial obstructions to the existence of Hamiltonian structures for fiberwise linear dynamics.