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**An index theory for periodic solutions of convex Hamiltonian systems.**
*(English)*
Zbl 0596.34023

Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/1, 395-423 (1986).

[For the entire collection see Zbl 0583.00018.]

This paper deals with the problem of finding, counting and describing periodic solutions of the 2n-dimensional system of ODEs: \(\dot x=JH'(x)\) where \(J=-J^*=J^{-1}\) defines a symplectic structure on \({\mathbb{R}}^{2n}\), and the Hamiltonian \(H: {\mathbb{R}}^{2n}\to {\mathbb{R}}\) is convex and \(C^ 3\). We study the fixed-energy problem, that is, we prescribe the energy level h, and we seek solutions x such that \(H(x(t))=h\) and \(x(t+T)=x(t)\) for all t. The hypersurface defined by the equation \(H(x)=h\) will henceforth be denoted by \(\Sigma\), and it will be assumed that it surrounds the origin. We associate with each periodic solution on \(\Sigma\) an index, which is a non-negative integer, and we relate this index with qualitative properties of the solution namely its length and stability. Here is a sample of the results we obtain: (a) set \(r=Inf\{| x| | \quad x\in \Sigma \}\quad and\quad R=Sup\{| x| | \quad x\in \Sigma \}.\) Then all periodic solutions x satisfy \(\int (Jx,\dot x)dt\geq \pi r^ 2\) (this is due to Croke and Weinstein) and there is at least one periodic solution with \(\int (Jx,\dot x)dt\leq \pi R^ 2;\) (b) assume now that \(\Sigma\) is (r,R)-pinched (this is a condition which is somewhat stronger than the preceding one; it means that \(\Sigma\) lies between two concentric balls with radii r and R, plus an analogous condition on the radii of curvature). If \(R/r<\sqrt{2}\), there is at least one periodic solution which is linearly stable. In (a) as in (b), the solution considered has index zero.

This paper deals with the problem of finding, counting and describing periodic solutions of the 2n-dimensional system of ODEs: \(\dot x=JH'(x)\) where \(J=-J^*=J^{-1}\) defines a symplectic structure on \({\mathbb{R}}^{2n}\), and the Hamiltonian \(H: {\mathbb{R}}^{2n}\to {\mathbb{R}}\) is convex and \(C^ 3\). We study the fixed-energy problem, that is, we prescribe the energy level h, and we seek solutions x such that \(H(x(t))=h\) and \(x(t+T)=x(t)\) for all t. The hypersurface defined by the equation \(H(x)=h\) will henceforth be denoted by \(\Sigma\), and it will be assumed that it surrounds the origin. We associate with each periodic solution on \(\Sigma\) an index, which is a non-negative integer, and we relate this index with qualitative properties of the solution namely its length and stability. Here is a sample of the results we obtain: (a) set \(r=Inf\{| x| | \quad x\in \Sigma \}\quad and\quad R=Sup\{| x| | \quad x\in \Sigma \}.\) Then all periodic solutions x satisfy \(\int (Jx,\dot x)dt\geq \pi r^ 2\) (this is due to Croke and Weinstein) and there is at least one periodic solution with \(\int (Jx,\dot x)dt\leq \pi R^ 2;\) (b) assume now that \(\Sigma\) is (r,R)-pinched (this is a condition which is somewhat stronger than the preceding one; it means that \(\Sigma\) lies between two concentric balls with radii r and R, plus an analogous condition on the radii of curvature). If \(R/r<\sqrt{2}\), there is at least one periodic solution which is linearly stable. In (a) as in (b), the solution considered has index zero.