An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

Abstract

We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal{M},\omega )$ is a symplectic manifold, $\mathfrak{L}$ is a distribution of Lagrangian subspaces in $\mathcal{M}$, $\mathcal{P}$ a Lagrangian submanifold of $\mathcal{M}$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal{M}$ and $\Gamma :[a,b]\to \mathcal{M}$ is an integral curve of the Hamiltonian flow $\Hf$ starting at $\mathcal{P}$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $\Gamma (b)$ is not $\mathcal{P}$-focal it is introduced the Maslov index $\maslov(Γ)$ of $\Gamma$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\maslov(Γ)$ is computed as a sort of {\em algebraic count} of the $\mathcal{P}$-focal points along $\Gamma$. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of $\Gamma$ is equal to the sum of $\maslov(Γ)$ and a {\em convexity term} of the Hamiltonian $H$ relative to the submanifold $\mathcal{P}$. When the result is applied to the case of the cotangent bundle $\mathcal{M}=T{M}^{\ast }$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac{1}{2}{g}^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.

34 pages, LaTeX2e, amsart class. This is the final version of the paper; it will appear in the Proceedings of the London Mathematical Society