Stability of equilibrium points for a Hamiltonian systems with one degree of freedom in one degenerate case
Hamiltonian systems have a wide application in theoretical physics and celestial mechanics. Some important topics in Hamiltonian theory are existence of quasi-periodic, periodic and stationary solutions, and stability of equilibrium points. The main theoretical results are combined in the Kolmogorov-Arnold-Moser theory. The authors consider one degree of freedom Hamiltonian systems such that the origin is an equilibrium solution and the Hamiltonian function \(H=H(q,p,t)\) is a real analytic function of \((q,p)\), \(2\pi\)-periodic in \(t\), in a neighborhood of the equilibrium point. Additionally, it is assumed that the Taylor series of \(H\) has the form \(H=H_2+H_3+\dots+H_j+\dots\), where \(H_j\) is an homogeneous polynomial of degree \(j\) in \((q,p)\). The so-called degenerate case occurs when the point \((0,0)\) is the equilibrium point of the above Hamiltonian system. The cases \(j=4,5,6\) are considered in detail. Some new results connected with the stability problem are found and discussed.
scientific article; zbMATH DE number 6955785