An Index Theorem for Non-Periodic Solutions of Hamiltonian Systems

We consider a Hamiltonian setup M, , H, L, , P, where M, isa symplectic manifold, L is a distribution of Lagrangian subspacesin M, P is a Lagrangian submanifold of M, H is a smooth time-dependentHamiltonian function on M, and :a,b] M is an integral curveof the Hamiltonian flow starting at P. We do not require any convexity property of the Hamiltonianfunction H. Under the assumption that (b) is not P-focal, weintroduce the Maslov index i_maslov of given in terms of thefirst relative homology group of the Lagrangian Grassmannian;under generic circumstances i_maslov() is computed as a sortof algebraic count of the P-focal points along . We prove thefollowing version of the Index Theorem: under suitable hypotheses,the Morse index of the Lagrangian action functional restrictedto suitable variations of is equal to the sum of i_maslov()and a convexity term of the Hamiltonian H relative to the submanifoldP. When the result is applied to the case of the cotangent bundleM = TM* of a semi-Riemannian manifold (M, g) and to the geodesicHamiltonian , we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesicswith variable endpoints in Riemannian geometry. 2000 MathematicalSubject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.