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Periodic solutions of Hamiltonian systems with superquadratic potential. (English) Zbl 0632.34036
The authors consider the Hamiltonian H(t,p.q), \[ H(t,p.q)=\sum^{n}_{i,j=1}a_{ij}(t,q)p_ ip_ j+\sum^{n}_{i=1}b_ i(t,q)p_ i+V(t,q), \] with suitable growth conditions on the coefficients. The important ones are (i) V(t,q) grows more than \(| q|^{\alpha}\) \((\alpha >2)\) at infinity, (ii) some plausible growth rates of \(a_{ij}(t,q)\) with \(q_ j\) in addition to \(\sum_{i,j}a_{ij}(t,p,q)p_ ip_ j>\nu (q)| p|^ 2\) \((\nu (q)>0)\), and (iii) relative growths of \(| a_{ij}|\), \(b^ 2_ j\) and \(| \partial b_ j/\partial q_ kq_ k|^ 2\) with respect to V(t,q) as \(q\to \infty\). It is shown that, in addition to the above, (1) if \(\partial H/\partial t=0\), the system has infinitely many nonconstant T-periodic solutions for every prescribed periodic T, and (2) if H(t,p.q) is T-periodic in t and satisfies suitable bounds on the second order differential coefficients with respect to p,q, then the system has at least a nonconstant T-periodic solution. In order to establish these results, the authors first demonstrate an abstract theorem on critical points for strongly indefinite functionals. As a prelude to this, they recall some notions on the critical point theory and the index theory.
Reviewer: N.D.Sengupta

MSC:
34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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