An Index Theorem for Non Periodic Solutions of Hamiltonian Systems
| dc.creator | Piccione, Paolo | |
| dc.creator | Tausk, Daniel Victor | |
| dc.date | 1999-11-08 | |
| dc.date | 2000-10-30 | |
| dc.date.accessioned | 2026-07-07T05:31:29Z | |
| dc.date.available | 2026-07-07T05:31:29Z | |
| dc.description | We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ 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 $Γ:[a,b]\to\mathcal M$ 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 $Γ(b)$ is not $\mathcal P$-focal it is introduced the Maslov index $\maslov(Γ)$ of $Γ$ 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 $Γ$. 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 $Γ$ 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 $\mathcal M=TM^*$ 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. | |
| dc.description | 34 pages, LaTeX2e, amsart class. This is the final version of the paper; it will appear in the Proceedings of the London Mathematical Society | |
| dc.identifier | https://arxiv.org/abs/math/9911047 | |
| dc.identifier | http://arxiv.org/abs/math/9911047 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79362 | |
| dc.subject | Differential Geometry | |
| dc.subject | Functional Analysis | |
| dc.subject | 34B24; 58E05; 58E10; 58F05; 70H20 | |
| dc.title | An Index Theorem for Non Periodic Solutions of Hamiltonian Systems | |
| dc.type | text |