An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

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 a Lagrangian submanifold of M, H is a smooth time dependent Hamiltonian function on M and Γ : [a, b] 7→ 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 it is introduced the Maslov index imaslov(Γ) 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 two versions of the Index Theorem. First, we show that, under suitable hypotheses, the Morse index of the Hamiltonian 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 . Moreover, we prove that the Maslov index imaslov(Γ) is characterized in terms of the negative eigenvalues of the Hamilton differential operator. When the results are 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) = 1 2 g(p, p), we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry. Date: October 1999. 1991 Mathematics Subject Classification. 34B24, 58E05, 58E10, 58F05, 70H20. The first author is partially sponsored by CNPq, the second author is sponsored by FAPESP.