On the theory of Hamiltonian systems with connections

Properties of Hamiltonian systems that are connected and related to the evolution of characteristic functions are considered: the bilinear form of canonical variables and the Poincare function. The relationship between the elementary Hamilton action and the full differential of the difference of these characteristic functions when the image point moves along a straight path is established. The conditions for the existence of normal configurations of systems and their first integrals depending on the structure of the Hamilton function are obtained.