Collision and non-collision solutions for a class of Keplerian-like dynamical systems. (English) Zbl 0832.70009

Based on one-to-one correspondence between periodic solutions of the second order Hamiltonian system \((*)\) \(q'' = - \nabla V(t,q (t))\) and the critical points of the functional \(f(u) = {1 \over 2} \int^T_0 |u'(t) |^2 dt - \int^T_0 V(t,u (t))dt\), where \(q(t) \in R^N\), \(u(t) \in H^1(S^1, R^N \backslash \{0\})\) is \(T\)- periodic in \(t\), and \(V \in C^1 (R \times R^N \backslash \{0\},R)\), \(V(t,x) \to - \infty\) as \(|x |\to 0\), the authors look for conditions on \(V\) which ensure the existence of a non-collisional (classical) solution of \((*)\). First of all, they prove that every generalized solution of an autonomous system has only finitely many collisions and then, extending this result to non-autonomous systems and using Morse index theory, provide a bound on the number of collisions. Finally, they prove the existence of a non-collisional solution of \((*)\) when \(V\) behaves near the origin like \(- |x |^\alpha\) with \(\alpha \in (1,2)\).
Reviewer: I.Mladenov (Sofia)


70F05 Two-body problems
70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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