## 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)

### MSC:

 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