Periodic solutions on a convex energy surface of a Hamiltonian system. II: A quantitative estimate for theorem by A. Weinstein concerning normal modes (Q1089965)

[For part I see ibid. 9, 213-222 (1986; Zbl 0609.58035).] The author studies the number of periodic solutions on a given compact strictly convex energy surface S of the Hamiltonian system (1) \(\dot z=JH'(z)\), with \(z\in {\mathbb{R}}^{2n}={\mathbb{C}}^ n\) and \(H\in C^ 2({\mathbb{R}}^{2n};{\mathbb{R}})\). The main result is the following. Let \(\omega =(\omega_ 1,...,\omega_ n)\) with \(0<\omega_ 1\leq \omega_ 2...\leq \omega_ n=1\), and \(Q_{\omega}=\{z\in {\mathbb{C}}^ n|\sum \omega_ j| z_ j|^ 2\leq 1\}\). Then there exists a number \(\rho =\rho (\omega)>1\) such that the following holds: if C is a compact strictly convex subset of \({\mathbb{C}}^ n\) with \(C^ 2\)-boundary S such that (i) \(r_ 1Q_{\omega}\subset C\subset r_ 2Q_{\omega}\) for some \(r_ 1,r_ 2>0\) with \(r_ 2<\rho r_ 1\), and (ii) S is an energy surface of H, then (1) has at least n distinct periodic solutions on S. The paper also gives an explicit construction for the number \(\rho\) (\(\omega)\). This theorem is related to the results of \textit{I. Ekeland} and \textit{J.-M. Lasry} [Ann. Math., II. Ser. 112, 283-319 (1980; Zbl 0449.70014)] but uses a somewhat different approach to distinguish geometrically different orbits, thus leading to partly independent results.