An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

Details An Index Theorem for Non Periodic Solutions of Hamiltonian Systems An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

1999-11-08

arxiv.org/abs/math/9911047v4

Mathematics

Differential Geometry

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

Scientific paper

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.

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