Periodic solutions on a convex energy surface of a Hamiltonian system (Q1086893)

Existence theorems of periodic solutions on a convex energy surface of a Hamiltonian system (1) \(\dot p=-H_ q\), \(\dot q=H_ p\), where \(p=(p_ 1,...,p_ n)\), \(q=(q_ 1,...,q_ n)\in R^ n\), \(H\in C^ 2(R^{2n},R)\) are established. Let \([s]_+=[s]_-=s\) for \(s\in Z\); \([s]_+=j+1\), \([s]_-=j\) for \(s\in (j,j+1)\) with \(j\in Z\), and let Q be the ellipsoid \(\{z\in C^ n: \sum^{n}_{j=1}\omega_ j| z_ j|^ 2\leq 1\}\), where \(z=(z_ 1,...,z_ n)\), \(z_ j=p_ j+iq_ j\), \(0<\omega_ 1\leq \omega_ 2\leq...\leq \omega_ n=1\). The main result of the paper is the following Theorem: Let C be a compact strictly convex subset of \(R^ n\) with \(C^ 2\) boundary S, \(K\geq 1\) and \(N\in Z\) with \(N>K\). For some \(h\in R\), \(H^{-1}(h)=S\) and H'(z)\(\neq 0\) for any \(z\in S\). Assume further that \(r,R\in R^+\) with \(R^ 2<N/Kr\) exist such that rB\(\subset C\subset RQ\), where B is closed unit ball in \(R^ n\). Then there exist at least [\(\sum^{n}_{j=1}[\omega_ jK]_-/(N- 1)]_+\) distinct periodic solutions of (1) on S.