On the covering of a Hill's region by solutions in systems with gyroscopic forces

Consider a Lagrangian system with the Lagrangian containing terms linear in velocity. By analogy with the systems in celestial mechanics, we call a bounded connected component of the possible motion area of such a system a Hill's region. Suppose that the energy level is fixed and the corresponding Hill's region is compact. We present sufficient conditions under which any point in the Hill's region can be connected with its boundary by a solution with the given energy. The result is illustrated by examples from mechanics.