Canonical transformations of linear Hamiltonian systems

\nIn this paper we consider the linear Hamiltonian systems of differential equations. We explore the normalization of a non-singular Hamiltonian matrix. We solve a system of matrix equations to find the generating function of the canonical transformation. In various cases we obtain the solution of the system of matrix equations. We get the solution of the algebraic matrix Riccati equation under certain conditions. Some properties of the Hamiltonian matrix have been proven. We get the normal form of a non-singular Hamiltonian matrix of order 4. We obtain the new method of normalization of the quadratic Hamiltonian. With this method we can investigate the stability of solutions of Hamiltonian systems.\n