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Dynamical systems with Newtonian type potentials. (English) Zbl 0692.34050

This paper is concerned with periodic solutions to the conservative dynamical system in \({\mathbb{R}}^ n\) given by \((*)q''+\text{grad} V(q)=0,\) where V(q) is a potential of Newtonian type (i.e. \(V(x)\approx -1/| x|^{\alpha}\) for \(\alpha\geq 1)\). The authors assume that V(x) is defined in a neighborhood of the origin and \(\lim_{| x| \to 0}V(x)=-\infty;\) they then look for periodic solutions which do not cross the origin. The authors’ main result is the existence of an origin- avoiding T-periodic solution to (*) provided that:
(1) V can be estimated via \(\psi_ 0(1/| x|)\leq -V(x)\leq \psi_ 1(1/| x|)\) for all x in a neighborhood of the origin where \(\psi_ 0,\psi_ 1: (0,\infty)\to [0,\infty)\) are non-constant and convex with \(\lim_{s\to 0}\psi_ i(s)=0\), and
(2) \(\phi_ 1(\psi_ 1,T)\leq \phi_ 0(\psi_ 0,T)\) where \(\phi_ 0\) gives a lower estimate of the Lagrangian integral on curves which meet the singularity, and \(\phi_ 1\) gives an upper estimate of the Lagrangian integral on the circular trajectories of minimum period T and speed of constant modulus, which lie on a suitable sphere centered at the origin.
Reviewer: W.J.Satzer jun.

MSC:

37-XX Dynamical systems and ergodic theory
34C25 Periodic solutions to ordinary differential equations
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