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 $\maslov(\Gamma)$ of $\Gamma$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\maslov(\Gamma)$ 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 $\maslov(\Gamma)$ 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 $\mathcal M=TM^*$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac12 g^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.