Properties of canonical transformations of linear Hamiltonian systems

Abstract

In this paper we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems have an important role in fluid mechanics and in statistical mechanics. Firstly, we prove some properties of canonical transformations of linear Hamiltonian systems. Further, we get the new method to find the generating function of a canonical transformation. We get a system of matrix equations for finding the generating function of a canonical transformation. In various cases we obtain the solution of the system. Then we use the system for normalization the Hamiltonian matrix. We apply the system of matrix equations to transform the Hamiltonian matrix from one normal form to another. Further, we get the solution of the matrix Riccati equation. This nonlinear equation has an important role in optimal control problems, multivariable and large-scale systems, scattering theory, estimation, detection and transportation. Finally, we transform the Hamiltonian from the complex form to the real form. An illustrative example for the proposed method is given. Therefore, we obtain the new method of normalization of the quadratic Hamiltonian. With this method we can investigate the stability of the solution of the Hamiltonian systems.