A hybrid numerical method for analysis of dynamics of the classical Hamiltonian systems

The numerical integration methods based on the forward and backward expansions of solutions in the Taylor series for some classical Hamiltonian systems are considered. The analytical representations of derivatives of the Hamiltonian are used for construction of the hybrid schemes of the approximate solutions of the Cauchy problem. The considered approach allows us also to study the solutions in the neighborhood of the singular points of the Hamiltonian. The efficiency of these hybrid implicit methods is illustrated on examples of numerical analysis of solutions for some Hamiltonian systems such as Toda and Henon-Heiles models, the system of Coulomb particles, and the three-body gravitational system on a line. A discrete time representation of the evolution of the three-body system on a line connected with constructing pair collision transition operators and Poincaré sections is discussed.