Perturbation results for a class of singular Hamiltonian systems. (English) Zbl 0742.58047

The authors consider the Hamiltonian system with \(n\) degrees of freedom \[ dp/dt=-\partial H/\partial q, dq/dt=\partial H/\partial p \] where the Hamiltonian \(H=H(t,p,q)\) is \(T\)-periodic in \(t\) and has a singularity at \(q=0\). The authors prove that under some stated conditions there exist a neighbourhood \(N\) of \(z\) and \(\varepsilon > 0\) such that for all \(| \varepsilon | < \varepsilon'\), \(-Jdz/dt=\nabla H(t,z)\) has at least \(\text{cat}(z)\) \(T\)-periodic solutions in \(N\). Where \(\text{cat}(.)\) denotes the Lyusternik-Shnirelman category, \(z=(p,q)\), and \(J\) is the symplectic matrix \(J(p,q)=(-q,p)\).
Reviewer: H.S.Nur (Fresno)


37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces