On the Constructive Algorithm for Stability Investigation of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in First-Order Resonance Case

Abstract

We consider a non-autonomous Hamiltonian system with two degrees of freedom, whose Hamiltonian function is a 2π-periodic function of time and is analytic in the neighborhood of an equilibrium point. It is assumed that the system exhibits a first-order resonance, i.e., the linearized system in the neighborhood of the equilibrium point has a unit multiplier of multiplicity two. The case of the general position is considered when the monodromy matrix is not reduced to the diagonal form, and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system. In this paper, a constructive algorithm for the rigorous-stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed by Markeev. The sufficient conditions for the instability of the equilibrium position, as well as the conditions for its formal stability and stability in the third approximation, are expressed in terms of the coefficients of the normalized map. Explicit formulas are obtained that allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of the symplectic map. The developed algorithm is used to solve the problem of the stability of the resonant rotation of a symmetric satellite.