## 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.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 70H05 Hamilton’s equations

Zbl 0583.00018