• DOI:10.1112/plms/83.2.351
• Corpus ID: 13435807

An Index Theorem for Non‐Periodic Solutions of Hamiltonian Systems

@article{Piccione1999AnIT,
  title={An Index Theorem for Non‐Periodic Solutions of Hamiltonian Systems},
  author={Paolo Piccione and Daniel V. Tausk},
  journal={Proceedings of the London Mathematical Society},
  year={1999},
  volume={83}
}

• P. Piccione, D. Tausk
• Published 8 November 1999
• Mathematics
• Proceedings of the London Mathematical Society

We consider a Hamiltonian setup M, ω, H, L, Γ, P, where M, ω is a symplectic manifold, L is a distribution of Lagrangian subspaces in M, P is a Lagrangian submanifold of M, H is a smooth time‐dependent Hamiltonian function on M, and Γ:[a,b] → M is an integral curve of the Hamiltonian flow H→ starting at P. We do not require any convexity property of the Hamiltonian function H. Under the assumption that Γ(b) is not P‐focal, we introduce the Maslov index imaslovΓ of Γ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances imaslov(Γ) is computed as a sort of algebraic count of the 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 imaslov(Γ) and a convexity term of the Hamiltonian H relative to the submanifold P. When the result is applied to the case of the cotangent bundle M = TM* of a semi‐Riemannian manifold (M, g) and to the geodesic Hamiltonian H(q,p)=12g−1(p,p) , we obtain a semi‐Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry. 2000 Mathematical Subject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.